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(1)Advanced stochastic processes: Part II Jan A. Van Casteren. Download free books at.

(2) Jan A. Van Casteren. Advanced stochastic processes Part II. ii Download free eBooks at bookboon.com.

(3) Advanced stochastic processes: Part II 2nd edition © 2015 Jan A. Van Casteren & bookboon.com ISBN 978-87-403-1116-7. iii Download free eBooks at bookboon.com.

(4) Advanced stochastic processes: Part II. Contents. Contents Contents. To See Part 1 download: Advanced stochastic processes: Part 1 Preface. i. Chapter 1i Preface 1. Stochastic processes: prerequisites 1. Conditional expectation 2 Chapter 1. Stochastic processes: prerequisites 19 2. Lemma of Borel-Cantelli 1. expectation 3. Conditional Stochastic processes and projective systems of measures 102 2. Lemma of Borel-Cantelli 4. A deﬁnition of Brownian motion 169 3. and processes projective systems of measures 10 5. Stochastic Martingalesprocesses and related 17 4. A deﬁnition of Brownian motion 16 Chapter 2. Renewal theory and Markov chains 35 5. Martingales and related processes 17 1. Renewal theory 35 Chapter 2. additional Renewal theory and Markov chains 35 2. Some comments on Markov processes 61 1. theory 35 3. Renewal More on Brownian motion 70 2. additional 61 4. Some Gaussian vectors.comments on Markov processes 76 3. on Brownian motion 70 5. More Radon-Nikodym Theorem 78 4. vectors. 76 6. Gaussian Some martingales 78 5. Radon-Nikodym Theorem 78 Chapter 3. An introduction to stochastic processes: Brownian motion, 6. Some martingales 78 Gaussian processes and martingales 89 Chapter 3. An introduction to stochastic processes: Brownian motion, 1. Gaussian processes 89 Gaussian and processes martingales 89 2. Brownian motionprocesses and related 98 1. Gaussian processes 3. Some results on Markov processes, on Feller semigroups and on the 89 2. Brownian motion and related processes 98 martingale problem 117 3. Some results on Markov processes, on Feller semigroups and on the 4. Martingales, submartingales, supermartingales and semimartingales 147 martingaleproperties problem of stochastic processes 117 5. Regularity 151 www.sylvania.com 4. Martingales, submartingales, supermartingales and semimartingales 147 6. Stochastic integrals, Itˆo’s formula 162 5. properties 151 7. Regularity Black-Scholes model of stochastic processes 188 6. Stochastic integrals, Itˆ o ’s formula 162 8. An Ornstein-Uhlenbeck process in higher dimensions 197 7. model theorem 188 9. Black-Scholes A version of Fernique’s 221 We do not reinvent 8. Ornstein-Uhlenbeck process in higher dimensions 197 10. An Miscellaneous 223 9. A version of Fernique’s theorem 221 the wheel we reinvent Index 237 10. Miscellaneous 223. light.. Chapter 243 Index 4. Stochastic diﬀerential equations 237 spectrum of 1. Solutions to stochastic diﬀerential equationsFascinating lighting offers an infinite243 possibilities: Innovative technologies and new Chapter 4. Stochastic diﬀerential theorem equations 243 2. A martingale representation 272 markets provide both opportunities and challenges. 1. to stochastic diﬀerential equationsAn environment in which your expertise 243 3. Solutions Girsanov transformation 277 is in high demand. Enjoy the supportive working 2. A martingale representation theorem 272atmosphere within our global group and benefit from Chapter 5. Some related results 295 3. Girsanov transformation 277international career paths. Implement sustainable ideas in close 1. Fourier transforms 295 cooperation with other specialists and contribute to Chapter 5. Some related results 295 2. Convergence of positive measures influencing our future. Come and join324 us in reinventing 1. Fourier transforms 295 light every day. iii 2. Convergence of positive measures 324 iii Light is OSRAM. iv Download free eBooks at bookboon.com. Click on the ad to read more.

(5) 1. Renewal theory 2. Some additional comments on Markov processes 3. More on Brownian motion 4. Gaussian vectors. Advanced stochastic processes: Part II 5. Radon-Nikodym Theorem 6. Some martingales. 35 61 70 76 Contents 78 78. Chapter 3. An introduction to stochastic processes: Brownian motion, Gaussian processes and martingales 1. Gaussian processes 2. Brownian motion and related processes 3. Some results on Markov processes, on Feller semigroups and on the martingale problem 4. Martingales, submartingales, supermartingales and semimartingales 5. Regularity properties of stochastic processes 6. Stochastic integrals, Itˆo’s formula 7. Black-Scholes model 8. An Ornstein-Uhlenbeck process in higher dimensions 9. A version of Fernique’s theorem 10. Miscellaneous. 117 147 151 162 188 197 221 223. Index. 237. 360° thinking. Chapter 4. Stochastic diﬀerential equations 1. Solutions to stochastic diﬀerential equations 2. A martingale representation theorem 3. Girsanov transformation. .. Chapter 5. Some related results 1. Fourier transforms 2. Convergence of positive measures. 89 89 98. 243 243 272 277 295 295 324. iii. 360° thinking. .. 360° thinking. .. Discover the truth at www.deloitte.ca/careers. © Deloitte & Touche LLP and affiliated entities.. Discover the truth at www.deloitte.ca/careers. Deloitte & Touche LLP and affiliated entities.. © Deloitte & Touche LLP and affiliated entities.. Discover the truth v at www.deloitte.ca/careers Click on the ad to read more Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities.. Dis.

(6) 7. Black-Scholes model 8. An Ornstein-Uhlenbeck process in higher dimensions 9. A version of Fernique’s theorem 10. Miscellaneous Advanced stochastic processes: Part II. 188 197 221 223 Contents. Index. 237. Chapter 4. Stochastic diﬀerential equations 1. Solutions to stochastic diﬀerential equations 2. A martingale representation theorem 3. Girsanov transformation. 243 243 272 277. Chapter 5. Some related results CONTENTS iv 1. Fourier transforms 2. Convergence of positive measures 3. A taste of ergodic theory iii 4. Projective limits of probability distributions 5. Uniform integrability 6. Stochastic processes 7. Markov processes 8. The Doob-Meyer decomposition via Komlos theorem Subjects for further research and presentations. 295 295 324 340 357 369 373 399 409 423. Bibliography. 425. Index. 433. We will turn your CV into an opportunity of a lifetime. Do you like cars? Would you like to be a part of a successful brand? We will appreciate and reward both your enthusiasm and talent. Send us your CV. You will be surprised where it can take you.. vi Download free eBooks at bookboon.com. Send us your CV on www.employerforlife.com. Click on the ad to read more.

(7) Advanced stochastic processes: Part II. Stochastic differential equations. CHAPTER 4. Stochastic diﬀerential equations Some pertinent topics in the present chapter consist of a discussion on martingale theory, and a few relevant results on stochastic diﬀerential equations in spaces of ﬁnite dimension. In particular unique weak solutions to stochastic differential equations give rise to strong Markov processes whose one-dimensional distributions are governed by the corresponding second order parabolic type diﬀerential equation. Essentially speaking this chapter is part of Chapter 1 in [146]. (The author is thankful to WSPC for the permission to include this text also in the present book.) In this chapter we discuss weak and strong solutions to stochastic diﬀerential equations. We also discuss a version of the Girsanov transformation. 1. Solutions to stochastic diﬀerential equations Basically, the material in this section is taken from Ikeda and Watanabe [61]. In Subsection 1.1 we begin with a discussion on strong solutions to stochastic diﬀerential equations, after that, in Subsection 1.2 we present a martingale characterization of Brownian motion. We also pay some attention to (local) exponential martingales: see Subsection 1.3. In Subsection 1.4 the notion of weak solutions is explained. However, ﬁrst we give a deﬁnition of Brownian motion which starts at a random position. 4.1. Definition. Let pΩ, F, Pq be a probability space with ﬁltration pFt qtě0 . A d-dimensional Brownian motion is a almost everywhere continuous adapted process tBptq “ pB1 ptq, . . . , Bd ptqq : t ě ` 0u ˘ such that for 0 ă t1 ă t2 ă ¨ ¨ ¨ ă d n tn ă 8 and for C any Borel subset of R the following equality holds: P rpB pt1 q ´ Bp0q, . . . , B ptn q ´ Bp0qq P Cs ż ż “ ¨ ¨ ¨ p0,d ptn ´ tn´1 , xn´1 , xn q ¨ ¨ ¨ p0,d pt2 ´ t1 , x1 , x2 q p0,d pt1 , 0, x1 q C. dx1 . . . dxn .. (4.1). This process is called a d-dimensional Brownian motion with initial `distribution ˘n`1 µ if for 0 ă t1 ă t2 ă ¨ ¨ ¨ ă tn ă 8 and every Borel subset of Rd the following equality holds: P rpBp0q, B pt1 q , . . . , B ptn qq P Cs ż ż “ ¨ ¨ ¨ p0,d ptn ´ tn´1 , xn´1 , xn q ¨ ¨ ¨ p0,d pt2 ´ t1 , x1 , x2 q p0,d pt1 , x0 , x1 q C. 243. 243 Download free eBooks at bookboon.com.

(8) Advanced stochastic processes: Part II 244. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. Stochastic differential equations. dµ px0 q dx1 . . . dxn .. (4.2). For the deﬁnition of p0,d pt, x, yq see formula (4.26). By deﬁnition a ﬁltration pFt qtě0 is an increasing family of σ-ﬁelds, i.e. 0 ď t1 ď t2 ă 8 implies Ft1 Ă Ft2 . The process of Brownian motion tBptq : t ě 0u is said to be adapted to the ﬁltration pFt qtě0 if for every t ě 0 the variable Bptq is Ft -measurable. It is assumed that the P-negligible sets belong to F0 . 1.1. Strong solutions to stochastic diﬀerential equations. In this section we discuss strong or pathwise solutions to stochastic diﬀerential equations. We also show that if the stochastic diﬀerential equation in (4.108) possesses unique pathwise solutions, then it has unique weak solutions. We begin with a formal deﬁnition. 4.2. Definition. The equation in (4.108) is said to have unique pathwise solutions, if for any Brownian motion tpBptq : t ě 0q , pΩ, F, Pqu and any pair of Rd -valued adapted processes tXptq : t ě 0u and tX 1 ptq : t ě 0u for which żt żt (4.3) Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds and 0 0 żt żt 1 1 (4.4) X ptq “ x ` σ ps, X psqq dBpsq ` b ps, X 1 psqq ds 0. 0. 1. it follows that Xptq “ X ptq P-almost surely for all t ě 0. If for any given Brownian motion pBptqqtě0 the process pXptqqtě0 is such that for P-almost all ω P Ω the equality żt żt Xpt, ωq “ x ` σ ps, Xps, ωqq dBps, ωq ` b ps, Xps, ωqq ds 0. 0. is true, then t ÞÑ Xptq is called a strong solution. Strong solutions are also called pathwise solutions. In order to facilitate the proof of Theorem 4.4 we insert the following lemma. 4.3. Lemma. Let γ be a positive real number. Then the following inequality holds: ˆ ´ 8 ¯ ¯2 1 ˙ ÿ a a 1 ´? γ n{2 1 ? ? ď . (4.5) γ ` γ ` 4 exp γ` γ`4 ´ 2 8 2 n! n“0 Since. ? ? ? γ ` γ ` 4 ď 2 γ ` 2, the inequality in (4.5) implies: ˙ ˆ 8 ÿ γ n{2 a 1 ? ď γ ` 2 exp pγ ` 1q ă 8. 2 n! n“0. We will use the ﬁniteness of the sum rather than the precise estimate.. 244 Download free eBooks at bookboon.com. (4.6).

(9) Advanced stochastic processes:TO Part II Stochastic differential 1. SOLUTIONS STOCHASTIC DIFFERENTIAL EQUATIONS 245 equations. Proof of Lemma 4.3. Let δ ą 0 be a positive number. Then we have by the Cauchy-Schwarz inequality ¸2 ˜ ¸2 ˜ 8 8 ÿ ÿ γ n{2 γ n{2 pδ ` γqn{2 ? ? “ pδ ` γqn{2 n! n! n“0 n“0 8 8 ÿ ÿ pδ ` γqn δ ` γ δ`γ γn “ e . ď (4.7) n pδ ` γq n“0 n! δ n“0 ´ ¯ a The choice δ “ 12 ´γ ` γpγ ` 4q yields the equalities ¯2 ¯2 a a 1 ´? δ`γ 1 ´? “ γ ` γ ` 4 ´ 1, and γ` γ`4 , δ`γ “ 4 δ 4 and so the result in (4.5) follows and completes the proof of Lemma 4.3. A version of the following result can be found in many books on stochastic diﬀerential equations: see e.g. [61, 107, 113].. I joined MITAS because I wanted real responsibili� I joined MITAS because I wanted real responsibili�. Real work International Internationa al opportunities �ree wo work or placements. �e Graduate Programme for Engineers and Geoscientists. Maersk.com/Mitas www.discovermitas.com. �e G for Engine. Ma. Month 16 I was a construction Mo supervisor ina const I was the North Sea super advising and the No he helping foremen advis ssolve problems Real work he helping fo International Internationa al opportunities �ree wo work or placements ssolve pr. 245 Download free eBooks at bookboon.com. Click on the ad to read more.

(10) Advanced stochastic processes: Part II 246. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. Stochastic differential equations. 4.4. Theorem. Let σj,k ps, xq and bj ps, xq, 1 ď j, k ď d be continuous functions defined on r0, 8q ˆ Rd such that for all t ą 0 there exists a constant Kptq with the property that d ÿ. j,k“1. 2. |σj,k ps, xq ´ σj,k ps, yq| `. d ÿ. j“1. |bj ps, xq ´ bj ps, yq|2 ď Kptq |x ´ y|2. (4.8). for all 0 ď s ď t, and all x, y P Rd . Fix x P Rd , and let pΩ, F, Pq be a probability space with a filtration pFt qtě0 . Moreover, let tBptq : t ě 0u be a Brownian motion on the filtered probability space pΩ, Ft , Pq. Then there exists“ an Rd -valued ‰ process tXptq : t ě 0u such that, for all 0 ă T ă 8, sup0ătďT E |Xptq|2 ă 8, and such that żt żt (4.9) Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds, t ě 0. 0. 0. This process is pathwise unique in the sense of Definition 4.2.. The techniques in the proof below are very similar to a method to prove the following version of Gronwall’s inequality: see e.g. ş[54]. Let f, g, h : r0, T s Ñ R t be continuous functions such that f ptq ď gptq ` 0 hpsqf psq ds, 0 ď t ď T . If h ě 0, then by induction with respect to k it follows that ´ş ´ş ¯j´1 ¯k t t ż ż k t t hpρq dρ hpρq dρ ÿ s s f ptq ď gptq ` gpsq ds ` hpsqf psq ds, pj ´ 1q! k! 0 j“1 0 and hence. żt. ˆż t. ˙ hpρq dρ ds.. f ptq ď gptq ` gpsq exp 0 s ` ˘˘ ` ˘ ` 2 d be the space of all continuous L2 Ω, F, P; Rd Let C r0, T s, L Ω, F, P; R valued functions supplied with the norm: ` “ ‰˘1{2 ` ˘ , X P L2 Ω, F, P; Rd . }X} “ sup E |Xptq|2 0ďtďT. ` ` ˘˘ ` ` ˘˘ Deﬁne the operator T : C r0, T s, L2 Ω, F, P; Rd Ñ C r0, T s, L2 Ω, F, P; Rd by the formula żt żt T Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds. 0. 0. Then ` the argumentation ` ˘˘in the`proof below ` shows that ˘˘ T is a mapping from 2 d 2 d C r0, T s, L Ω, F, P; R to C r0, T s, L Ω, F, P; R indeed, and that T has a unique ﬁxed point X which is a pathwise solution to the equation in (4.9). Proof. Existence. Fix 0 ă T ă 8. Put X0 psq “ x, 0 ď s ď t, and, for n ě 1, 0 ă t ď T , żt żt (4.10) Xn`1 ptq “ x ` b ps, Xn psqq ds ` σ ps, Xn psqq dBpsq. 0. 0. 246 Download free eBooks at bookboon.com.

(11) Advanced stochastic processes: Part II. Stochastic differential equations. 1. SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS. By (4.10) we see, for n ě 1 and 0 ă t ď T , żt Xn`1 ptq ´ Xn ptq “ pb ps, Xn psqq ´ b ps, Xn´1 psqqq ds 0 żt ` pσ ps, Xn psqq ´ σ ps, Xn´1 psqqq dBpsq.. 247. (4.11). 0. By assumption there exists functions s ÞÑ Kj psq and s ÞÑ Kij psq, 0 ď s ď T , such that for |bj ps, yq ´ bj ps, xq| ď Kj psq |y ´ x| , 0 ď s ď T, x, y P Rd , and. (4.12). (4.13) |σij ps, yq ´ σij ps, xq| ď Ki,j psq |y ´ x| , 0 ď s ď T, x, y P Rd , şT and such that 0 pKj psq2 ` Ki,j psq2 q ds ă 8 for 0 ď 1 ď i, j ď d. Let the ř ř function Kpsq ě 0 be such that Kpsq2 “ dj“1 Kj psq2 ` di“1 max1ďjďd Kij psq2 . şT Then 0 Kpsq2 ds ă 8. Moreover, for n ě 1 and 0 ď t ď T we infer, by using (4.11). (4.12) and (4.13), by the definition of Kpsq, and by standard properties of stochastic integrals relative to Brownian motion, the following inequality: żt “ ‰ ‰ “ 2 (4.14) E |Xn`1 ptq ´ Xn ptq| ď 2 Kpsq2 E |Xn psq ´ Xn´1 psq|2 ds. 0. In` order to ˘obtain (4.14) we also used an inequality of the form p|a| ` |b|q2 ď 2 |a|2 ` |b|2 , a, b P Rd . The proofs of (4.15) and (4.18) require equalities of the form ´ş ¯j t 2 ż ź ż j Kpρq dρ s , j P N, j ě 1. K psi q2 ds1 . . . dsj “ j! săs1 ă¨¨¨ăsj ăt i“1 By employing induction the inequality in (4.14) yields, for 1 ď j ď n and for 0 ď t ď T , the inequality: “ ‰ E |Xn`1 ptq ´ Xn ptq|2 ´ ¯j´1 ż t şt Kpρq2 dρ “ ‰ s Kpsq2 E |Xn´j`1 psq ´ Xn´j psq|2 ds. ď 2j (4.15) pj ´ 1q! 0. Since X0 psq “ x the equality in (4.10) for n “ 1 yields żs żs b pρ, xq dρ ` σ pρ, xq dBpρq, X1 psq ´ X0 psq “ 0. 0. and hence, for 0 ď s ď T , “. 2. E |X1 psq ´ X0 psq|. ‰. ¸ ˜ˇż ˇ2 d żs ÿ ˇ s ˇ 2 |σij pρ, xq| dρ . (4.16) ď 2 ˇˇ b pρ, xq dρˇˇ ` 0. i,j“1. 0. Let Aps, xq ě 0 be such that ˇż τ ˇ2 d żs ÿ ˇ ˇ 2 ˇ ˇ |σij pρ, xq|2 dρ. Aps, xq “ sup ˇ b pρ, xq dρˇ ` 0ăτ ďs 0. i,j“1 0. 247 Download free eBooks at bookboon.com. (4.17).

(12) Advanced stochastic processes: Part II 248 4. STOCHASTIC DIFFERENTIAL EQUATIONS Stochastic differential equations. Then (4.17) together with (4.15) with j “ n yields ´ ¯n´1 ż t şt Kpρq2 dρ “ “ ‰ ‰ s E |Xn`1 ptq ´ Xn ptq|2 ď 2n Kpsq2 E |X1 psq ´ X0 psq|2 ds pn ´ 1q! 0 ´ş ¯n´1 ż t t Kpρq2 dρ s Kpsq2 Aps, xq2 ds ď 2n`1 pn ´ 1q! 0 ´ ¯n´1 ż t şt Kpρq2 dρ s ď 2n`1 Apt, xq2 Kpsq2 ds pn ´ 1q! 0 ż ż ź n n`1 2 “ 2 Apt, xq K psj q2 ds1 . . . dsn 0ăs1 ă¨¨¨ăsn ăt. “ 2n`1 Apt, xq2. ´ş. ¯n t 2 Kpsq ds 0. j“1. . (4.18) n! şt From Lemma (4.3) and inequality (4.6) with γ “ 2 0 Kpsq2 ds we infer: d żt 8 ÿ şt ` “ ‰˘ 1 1{2 2 2 ď 2Apt, xq Kpsq2 ds ` 1 e 0 Kpsq ds` 2 . E |Xn`1 ptq ´ Xn ptq| 0. n“0. (4.19) From (4.19) it easily there exists an adapted R -valued process ` follows that ˘ pXptqq0ďtďT in L2 Ω, FT , P; Rd such that “ ‰ (4.20) lim E |Xn ptq ´ Xptq|2 “ 0. d. nÑ8. From (4.19) it also follows that this convergence also holds P-almost surely. The latter can be seen as follows. Fix η ą 0. Then the probability of the event tlim supnÑ8 |Xn ptq ´ Xptq| ą ηu can be estimated as follows: « ﬀ ȷ „ 8 ď t|Xn ptq ´ Xptq| ą ηu P lim sup |Xn ptq ´ Xptq| ą η ď inf P nÑ8. mPN. ď inf P mPN. ď inf P mPN. n“m. «. 8 ď. n1 ąn2 ěm. «#. 1 ď inf E mPN η. «. 8 ÿ. n“m 8 ÿ. n“m. ﬀ. t|Xn1 ptq ´ Xn2 ptq| ą ηu. |Xn`1 ptq ´ Xn ptq| ą η ﬀ. +ﬀ. |Xn`1 ptq ´ Xn ptq|. 8 1 ÿ E r|Xn`1 ptq ´ Xn ptq|s “ 0. (4.21) ď inf mPN η n“m. The ﬁnal equality is a consequence of Lemma 4.3 together with (4.19) and the ` “ ‰˘1{2 . Since η ą 0 is inequality E r|Xn`1 ptq ´ Xn ptq|s ď E |Xn`1 ptq ´ Xn ptq|2 arbitrary in (4.21) we infer that limnÑ8 Xn ptq “ Xptq (P-almost surely). This. 248 Download free eBooks at bookboon.com.

(13) Advanced stochastic processes:TO PartSTOCHASTIC II Stochastic differential 1. SOLUTIONS DIFFERENTIAL EQUATIONS 249 equations. P-almost sure convergence (as n Ñ 8) also implies that we may take pointwise limits in (4.10) to obtain: żt żt Xptq “ x ` b ps, Xpsqq ds ` σ ps, Xpsqq dBpsq. (4.22) 0. 0. The equality in (4.22) shows the existence of pathwise or strong solutions to the equation in (4.9). Uniqueness. Let pX1 ptqq0ďtďT and pX2 ptqq0ďtďT be two solutions to the stochastic diﬀerential equation in (4.9). By using a stopping time argument we may assume that sup0ďsďT |X2 psq ´ X1 psq| is P-almost surely bounded. Then żt X2 ptq ´ X1 ptq “ pb ps, X2 psqq ´ b ps, X1 psqqq ds 0 żt (4.23) ` pσ ps, X2 psqq ´ σ ps, X1 psqqq dBpsq. 0. As in the proof of (4.15) with j “ n and (4.18) it then follows that ´ ¯n´1 ż t şt Kpρq2 dρ “ ‰ ‰ “ s Kpsq2 E |X2 psq ´ X1 psq|2 ds E |X2 ptq ´ X1 ptq|2 ď 2n pn ´ 1q! 0 ´ş ¯n t 2 Kpρq dρ “ ‰ 0 . (4.24) ď 2n sup E |X2 psq ´ X1 psq|2 n! 0ăsăt Since the right-hand side of (4.24) tends to 0 as n Ñ 8 we see that X2 ptq “ X1 ptq P-almost surely. So uniqueness follows. . The proof of Theorem 4.4 is complete now.. 1.2. A martingale characterization of Brownian motion. The following result we owe to L´evy. 4.5. Theorem. Let pΩ, F, Pq be a probability space with filtration (or reference system) pFt qtě0 . Suppose F is the σ-algebra generated by Ytě0 Ft augmented with the P-zero sets, and suppose Ft is continuous from the right: Ft “ Xsąt Fs for all t ě 0. Let tM ptq “ pM1 ptq, . . . , Md ptqq : t ě 0u be an Rd -valued local P-almost surely continuous martingale with the property that the quadratic covariation processes t ÞÑ ⟨Mi , Mj ⟩ ptq satisfy ⟨Mi , Mj ⟩ ptq “ δi,j t,. 1 ď i, j ď d.. (4.25). Then tM ptq : t ě 0u is d-dimensional Brownian motion with initial distribution given by µpBq “ P rM p0q P Bs, B P BRd , the Borel field of Rd . It follows that the ﬁnite-dimensional distributions of the process t ÞÑ M ptq are given by: P rM pt1 q P B1 , . . . , M ptn q P Bn s ż ż ˆż ... p0,d ptn ´ tn´1 , xn´1 , xn q ¨ ¨ ¨ p0,d pt2 ´ t1 , x1 , x2 q p0,d pt1 , x, x1 q “ B1. Bn. 249 Download free eBooks at bookboon.com.

(14) Advanced stochastic processes: Part II 250. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. Stochastic differential equations. ˙. dxn ¨ ¨ ¨ dx1 dµpxq. Here p0,d pt, x, yq is the classical Gaussian kernel: ¸ ˜ 1 |x ´ y|2 p0,d pt, x, yq “ `? ˘d exp ´ . 2t 2πt. (4.26). 4.6. Remark. There is even a nicer result which says the following. Let X be a continuous Rd -valued process with stationary independent increments. Then, 2 there exist unique b P Rd and Σ P Rd such that X ptq´X p0q is a pb, Σq-Brownian motion. This means that Xptq is a Gaussian (or multivariate normal) vector such that E rXptqs “ bt and E rpXj1 ptq ´ bj1 tq pXj2 ptq ´ bj2 tqs “ tΣj1 ,j2 .. For the one-dimensional case the reader is referred to Breiman [29]. For the higher dimensional case, see, e.g., Lowther [89].. 250 Download free eBooks at bookboon.com. Click on the ad to read more.

(15) Advanced stochastic processes:TO PartSTOCHASTIC II Stochastic differential 1. SOLUTIONS DIFFERENTIAL EQUATIONS 251 equations. Proof of Theorem 4.5. Let ξ P Rd be arbitrary. First we show that it suﬃces to establish the equality: ˇ ‰ “ 2 1 (4.27) E e´i⟨ξ,M ptq´M psq⟩ ˇ Fs “ e´ 2 |ξ| pt´sq , t ą s ě 0.. For suppose that‰ (4.27) is true for all ξ P Rd . Observe that (4.27) implies “ ´i⟨ξ,M 2 1 ptq´M psq⟩ E e “ e´ 2 |ξ| pt´sq . Then, by standard approximation arguments, it follows that the variable M ptq ´ M psq is P-independent of Fs . In other words the process t ÞÑ M ptq possesses independent increments. Since the Fourier transform of the function y ÞÑ p0,d pt ´ s, 0, yq is given by ż 2 1 e´i⟨ξ,y⟩ p0,d pt ´ s, 0, yq dy “ e´ 2 |ξ| pt´sq Rd. it also follows that the distribution of M ptq ´ M psq is given by ż P rM ptq ´ M psq P Bs “ p0,d pt ´ s, 0, yq dy.. (4.28). B. Moreover, for 0 ă t1 ă ¨ ¨ ¨ ă tn we also have. P rM p0q P B0 , M pt1 q ´ M p0q P B1 , . . . , M ptn q ´ M ptn´1 q P Bn s. “ P rM p0q P B0 s P rM pt1 q ´ M p0q P B1 s ¨ ¨ ¨ P rM ptn q ´ M ptn´1 q P Bn s ż ż ż “ ¨¨¨ p0,d pt1 , 0, y1 q ¨ ¨ ¨ p0,d ptn ´ tn´1 , 0, yn q dµ py0 q dy1 ¨ ¨ ¨ dyn . B0. B1. Bn. Here B0 , . . . , Bn are Borel subsets of Rd . Hence, if B is a Borel subset of Rd ˆ ¨ ¨ ¨ ˆ Rd , then it follows that looooooomooooooon n`1times. P rpM p0q, M pt1 q ´ M p0q, . . . , M ptn q ´ M ptn´1 qq P Bs ż ż “ ¨ ¨ ¨ p0,d pt1 , 0, y1 q ¨ ¨ ¨ p0,d ptn ´ tn´1 , 0, yn q dµ py0 q dy1 ¨ ¨ ¨ dyn .. (4.29). B. Next we compute the joint distribution of pM p0q, M pt1 q , . . . , M ptn qq by employing (4.29). Deﬁne the linear map ℓ : Rd ˆ ¨ ¨ ¨ ˆ Rd Ñ Rd ˆ ¨ ¨ ¨ ˆ Rd by ℓ px0 , x1 , . . . , xn q “ px0 , x1 ´ x0 , x2 ´ x1 , . . . , xn ´ xn´1 q .. Let B be a Borel subset of Rd ˆ ¨ ¨ ¨ ˆ Rd . By (4.29) we get P rpM p0q, . . . , M ptn qq P Bs. “ P rℓ pM p0q, . . . , M ptn qq P ℓ pBqs. “ P rpM p0q, M pt1 q ´ M p0q, . . . , M ptn q ´ M ptn´1 qq P ℓ pBqs ż ż “ . . . p0,d pt1 , 0, y1 q ¨ ¨ ¨ p0,d ptn ´ tn´1 , 0, yn q dµ py0 q dy1 ¨ ¨ ¨ dyn ℓpBq. (change of variables: py0 , y1 , . . . , yn q “ ℓ px0 , x1 , . . . , xn q) ż ż “ ¨ ¨ ¨ p0,d pt1 , x0 , x1 q ¨ ¨ ¨ p0,d ptn ´ tn´1 , xn´1 , xn q dµ px0 q dx1 ¨ ¨ ¨ dxn . (4.30) B. 251 Download free eBooks at bookboon.com.

(16) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 252 4. STOCHASTIC. In order to complete the proof of Theorem 4.5 from equality (4.30) it follows that it is suﬃcient to establish the equality in (4.27). Therefore, ﬁx ξ P Rd and t ą s ě 0. An application of Itˆo’s lemma to the function x ÞÑ e´i⟨ξ,x⟩ yields e´i⟨ξ,M ptq⟩ ´ e´i⟨ξ,M psq⟩ żt żt d d ÿ 1 ÿ ´i⟨ξ,M pτ q⟩ “ ´i ξj e dMj pτ q ´ ξj ξk e´i⟨ξ,M pτ q⟩ d ⟨Mj , Mk ⟩ pτ q 2 s s j“1 j,k“1. (formula (4.25)) żt ż d ÿ 1 2 t ´i⟨ξ,M pτ q⟩ ´i⟨ξ,M pτ q⟩ ξj e dMj pτ q ´ |ξ| e dτ. “ ´i 2 s s j“1. (4.31). Hence, from (4.31) it follows that. e´i⟨ξ,M ptq´M psq⟩ ´ 1 (4.32) żt ż d ÿ 1 2 t ´i⟨ξ,M pτ q´M psq⟩ ´i⟨ξ,M pτ q´M psq⟩ “ ´i ξj e dMj pτ q ´ |ξ| e dτ. 2 s s j“1. Since the processes t ÞÑ. żt s. e´i⟨ξ,M pτ q´M psq⟩ dMj pτ q, t ě s, 1 ď j ď d,. are local martingales, we infer by (possibly) using a stopping time argument that ż “ ´⟨ξ,M ptq´M psq⟩ ˇ ‰ 1 2 t “ ´i⟨ξ,M pτ q´M psq⟩ ˇˇ ‰ ˇ E e Fs “ 1 ´ |ξ| Fs . E e (4.33) 2 s Next, let vptq, t ě s, be given by żt ˇ ‰ “ vptq “ E e´i⟨ξ,M pτ q´M psq⟩ ˇ Fs dτ. s. Then vpsq “ 0, and (4.33) implies. v 1 ptq `. 1 2 |ξ| vptq “ 1. 2. From (4.34) we infer ˙ ¯ ˆ1 2 2 1 1 d ´ 1 pt´sq|ξ|2 2 1 e2 |ξ| vptq ` v ptq e 2 pt´sq|ξ| “ e 2 pt´sq|ξ| . vptq “ dt 2 The equality in (4.35) implies: 1. 2. e 2 pt´sq|ξ| vptq ´ vpsq “ and thus we see. (4.34). (4.35). ¯ 2 ´ 1 pt´sq|ξ|2 2 e ´ 1 , |ξ|2. 2 2 1 1 1 v 1 ptq ` vpsqe´ 2 pt´sq|ξ| “ e´ 2 pt´sq|ξ| 2 Since vpsq “ 0 (4.36) results in ˇ ‰ “ 2 1 E e´i⟨ξ,M pτ q´M psq⟩ ˇ Fs “ v 1 ptq “ e´ 2 pt´sq|ξ| .. 252 Download free eBooks at bookboon.com. (4.36). (4.37).

(17) Advanced stochastic processes:TO PartSTOCHASTIC II Stochastic differential 1. SOLUTIONS DIFFERENTIAL EQUATIONS 253 equations. The equality in (4.37) is the same as the one in (4.27). By the above arguments this completes the proof of Theorem 4.5. As a corollary to Theorem 4.5 we get the following result due to L´evy. 4.7. Corollary. Let tM ptq : t ě 0u be a continuous local martingale in R such that the process t ÞÑ M ptq2 ´ t is a local martingale as well. Then the process tM ptq : t ě 0u is a Brownian motion with initial distribution given by µpBq “ P rM p0q P Bs, B P BR . Proof. Since M ptq2 ´ t is a local martingale, it follows that the quadratic variation process t ÞÑ ⟨M, M ⟩ ptq satisfies ⟨M, M ⟩ ptq “ t, t ě 0. So the result in Corollary 4.7 follows from Theorem 4.5. The following result contains a d-dimensional version of Corollary 4.7. 4.8. Theorem. Let tM ptq “ pM1 ptq, . . . , Md1 ptqq : t ě 0u be a continuous local martingale with covariation process given by żt ⟨Mj , Mk ⟩ ptq “ Φj,k psqds, 1 ď j, k ď d1 . (4.38) 0. ˚ Let the d1 ˆ d-matrix process tχptq : t ě 0u be such şt that χptqΦptqχptq “ I, where I is the d ˆ d identity matrix. Put Bptq “ 0 χpsq dM psq. This integral should be interpreted in Itˆo sense. Then the process t ÞÑ Bptq is d-dimensional Brownian motion. Put Ψptq “ Φptqχptq˚ , andş suppose that Ψptqχptq “ I, the t d1 ˆ d1 identity matrix. Then M ptq ´ M p0q “ 0 Ψpsq dBpsq.. 4.9. Remark. Since. χptq pΦptqχptq˚ χptq ´ Iq “ pχptqΦptqχptq˚ ´ Iq χptq “ 0. we see that the second equality in Ψptqχptq “ Φptqχptq˚ χptq “ I is only possible if we assume d “ d1 . Of course here we take the dimensions of the null and range space of the matrix χptq into account. Proof of Theorem 4.8. Fix 1 ď i, j ď d. We shall calculate the quadratic covariation process ⟩ ⟨ 1 ż d d1 ż p¨q p¨q ÿ ÿ pχpsqqi,k dMk psq, pχpsqqj,l dMl psq ptq ⟨Bi , Bj ⟩ ptq “ k“1. “ “. d1 ÿ d1 ż t ÿ. k“1 l“1 0 żt 0. 0. l“1 0. pχpsqqi,k pχpsqqj,l Φpsqk,l ds. pχpsqΦpsqχpsq˚ qi,j ds “ tδi,j .. (4.39). From Theorem 4.5 and (4.39) we see that the process t ÞÑ Bptq is a Brownian motion. This proves the first part of Theorem 4.8. Next we calculate żt żt żt Ψpsq dBpsq “ Ψpsqχpsq dM psq “ dM psq “ M ptq ´ M p0q, (4.40) 0. 0. 0. 253 Download free eBooks at bookboon.com.

(18) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 254 4. STOCHASTIC. which completes the proof of Theorem 4.8.. . 1.3. Exponential local martingales. Let t ÞÑ N ptq, 0 ď t ď T , be a continuous (local) martingale with variation process t ÞÑ ⟨N, N ⟩ ptq, 0 ď t ď T . In subsection we discuss local martingales of the form t ÞÑ e´Zptq “ şt this 1 ` 0 e´Zpsq dN psq, t ě 0, where Zptq “ N ptq ` 12 ⟨N, N ⟩ ptq. Such processes are called exponential local martingales. The following proposition serves as a preparation for Proposition 4.12. It also has some interest of its own. 4.10. Proposition. Let pΩ, F, Pq be a probability space with a filtration F0ďtďT , and let M “ pM ptqq0ďtďT and N “ pN ptqq0ďtďT be two local martingales “ ´Zptqwith ‰ 1 M p0q “ N p0q “ 0. Put Zptq “ N ptq` 2 ⟨N, N ⟩ ptq, and assume that E e “ 1 for all 0 ď t ď T . Then the following assertions are true.. (a) The process t ÞÑ e´Zptq , 0 ď t ď T , is a martingale; (b) The process t ÞÑ e´Zptq pM ptq ` ⟨N, M ⟩ ptqq is a local martingale; (c) The process t ÞÑ M ptq ` ⟨N, M ⟩ ptq is a local martingale relative to the supplied filtration pFt q0ďtďT “ ´ZpT ‰ with the measure QN : FT Ñ r0, 1s defined q 1A , A P FT . by QN pAq “ E e. no.1. Sw. ed. en. nine years in a row. STUDY AT A TOP RANKED INTERNATIONAL BUSINESS SCHOOL Reach your full potential at the Stockholm School of Economics, in one of the most innovative cities in the world. The School is ranked by the Financial Times as the number one business school in the Nordic and Baltic countries.. Stockholm. Visit us at www.hhs.se. 254 Download free eBooks at bookboon.com. Click on the ad to read more.

(19) Advanced stochastic processes:TO PartSTOCHASTIC II Stochastic differential 1. SOLUTIONS DIFFERENTIAL EQUATIONS 255 equations. The measure QN can“be called‰ a risk neutral measure. Observe that, by assertion (a), QN pAq “ E e´Zptq 1A whenever A belongs to Ft with 0 ď t ď T . Let τn be the stopping time defined by * " 1 τn “ inf s ą 0 : |N ps ^ T q| ` ⟨N, N ⟩ ps ^ T q ą n , 2. and set Zn ptq “ Z pt ^ τn q. Then the t ÞÑ e´Zn ptq , 0 ď t ď T , n P N, “ ´Zprocesses ‰ n ptq are martingales. It follows that E e “ 1, for all 0 ď t ď T , and for all n P N. By Fatou’s lemma we infer that ı ” “ ‰ ‰ “ (4.41) E e´Zptq “ E lim e´Zn ptq ď lim inf E e´Zn ptq ď 1. nÑ8. nÑ8. In fact we have a stronger result. It says that an exponential local martingale is a submartingale. 4.11. Theorem. Let the process t ÞÑ e´Zptq , 0 ď t ď T , be a continuous local“ martingale. In general, this process is a submartingale. Consequently, if ‰ E e´Zptq “ 1 for all 0 ď t ď T , then the process t ÞÑ e´Zptq , 0 ď t ď T , is a martingale. Proof. This result can be seen as follows. Let 0 ď t1 ă t2 , and choose the sequence of stopping times pτn qnPN as above. Then, for A P Ft1 ^τm , we have ı ” “ ‰ E e´Zpt2 q 1A “ E lim e´Zpt2 ^τn q 1A nÑ8 “ ‰ “ ‰ ď lim inf E e´Zpt2 ^τn q 1A “ E e´Zpt1 ^τm q 1A (4.42) nÑ8. From (4.42) it follows that: ˇ “ ‰ E e´Zpt2 q ˇ Ft1 ^τm ď e´Zpt1 ^τm q .. Since the event tτm ą t1 u belongs to Ft1 ^τm , from (4.43) we infer ˇ ˇ “ ‰ “ ‰ E e´Zpt2 q 1tτm ąt1 u ˇ Ft1 “ E e´Zpt2 q 1tτm ąt1 u ˇ Ft1 ^τm. ď e´Zpt1 ^τm q 1tτm ąt1 u “ e´Zpt1 q 1tτm ąt1 u .. (4.43). (4.44). The first equality in (4.44) is a consequence of the fact that, if an event A belongs to Ft1 , then A X tτm ą t1 u belongs to Ft1 ^τm . In the left-hand side and the far right-hand side of (4.44) we let m Ñ 8 to obtain ˇ “ ‰ E e´Zpt2 q ˇ Ft1 ď e´Zpt1 q , P-almost surely. (4.45) The inequality in (4.45) shows that‰ the process t ÞÑ e´Zptq is a submartingale. “ ´Zpt ‰ “ 2q If 0 ď t1 ă t2 ď T , and if E e “ E e´Zpt1 q , then (4.45) implies that ˇ ‰ “ (4.46) E e´Zpt2 q ˇ Ft1 “ e´Zpt1 q , P-almost surely. This completes the proof of Theorem 4.11.. . Proof of 4.10. (a) An application of Itˆo’s formula and employing the equality ⟨Z, Z⟩ ptq “ ⟨N, N ⟩ ptq yields: e´Zptq. 255 Download free eBooks at bookboon.com.

(20) Advanced stochastic processes: Part II 256. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. żt. Stochastic differential equations. ż 1 t ´Zpρq “e ´ e dZpρq ` e d ⟨Z, Z⟩ pρq 2 0 0 ż ż żt 1 t ´Zpρq 1 t ´Zpρq ´Zp0q ´Zpρq “e ´ e dN pρq ´ e d ⟨N, N ⟩ pρq ` e d ⟨N, N ⟩ pρq 2 0 2 0 0 żt ´Zp0q ´ e´Zpρq dN pρq. (4.47) “e ´Zp0q. ´Zpρq. 0. From the equalities in (4.47) it follows that the process t ÞÑ‰e´Zptq , 0 ď t ď T , is a “ ´Zptq local martingale. In view of the assumption that E e “ 1 for all 0 ď t ď T it follows that the process in (a) is a genuine martingale: see Theorem 4.11. (b) Again we apply Itˆo’s lemma, now to the function px, yq ÞÑ e´x y. Then we obtain: e´Zptq pM ptq ` ⟨N, M ⟩ ptqq żt żt ´Zpρq pM pρq ` ⟨N, M ⟩ pρqq dZpρq ` e´Zpρq pdM pρq ` d ⟨N, M ⟩ pρqq “´ e 0 0 żt 1 ` e´Zpρq pM pρq ` ⟨N, M ⟩ pρqq d ⟨Z, Z⟩ pρq 2 0 żt (4.48) ´ e´Zpρq d ⟨Z, M ` ⟨N, M ⟩⟩ pρq. 0. By applying the equalities ⟨Z, Z⟩ “ ⟨N, N ⟩ and ⟨Z, M ` ⟨N, M ⟩⟩ “ ⟨N, M ⟩ to the equality in (4.48) we obtain e´Zptq pM ptq ` ⟨N, M ⟩ ptqq żt “ ´ e´Zpρq pM pρq ` ⟨N, M ⟩ pρqq dN pρq 0 ż 1 t ´Zpρq ´ e pM pρq ` ⟨N, M ⟩ pρqq d ⟨N, N ⟩ pρq 2 0 żt ` e´Zpρq pdM pρq ` d ⟨N, M ⟩ pρqq 0 ż żt 1 t ´Zpρq ` e pM pρq ` ⟨N, M ⟩ pρqq d ⟨N, N ⟩ pρq ´ e´Zpρq d ⟨N, M ⟩ pρq 2 0 0 żt żt “ ´ e´Zpρq pM pρq ` ⟨N, M ⟩ pρqq dN pρq ` e´Zpρq dM pρq. (4.49) 0. 0. Being the sum of two stochastic integrals with respect to (local) martingales the equality in (4.49) implies that the process in (b) is a local martingale. (c) By using a stopping time argument we may and do assume that the process t ÞÑ M ptq ` ⟨N, M ⟩ ptq is bounded and so it belongs to L1 pΩ, FT , QN q. Let 0 ď t1 ă t2 ď T , and put ˇ ‰ “ Y pt1 q “ EQN M pt2 q ` ⟨N, M ⟩ pt2 q ˇ Ft1 . 256 Download free eBooks at bookboon.com.

(21) Advanced stochastic processes:TO Part II Stochastic differential 1. SOLUTIONS STOCHASTIC DIFFERENTIAL EQUATIONS 257 equations. Then the stochastic variable Y pt1 q is Ft1 -measurable and, for all bounded Ft1 measurable variables G we have ‰ “ ‰ “ (4.50) E e´ZpT q pM pt2 q ` ⟨N, M ⟩ pt2 qq G “ E e´ZpT q Y pt1 q G .. Since the process t ÞÑ e´Zptq , 0 ď t ď T , is a P martingale, the equality in (4.50) implies: “ ‰ “ ‰ E e´Zpt2 q pM pt2 q ` ⟨N, M ⟩ pt2 qq G “ E e´Zpt1 q Y pt1 q G . (4.51). From assertion (b) together with our stopping time argument we see that the process t ÞÑ e´Zptq pM ptq ` ⟨N, M ⟩ ptqq is a P-martingale. From (4.51) we then infer: ‰ “ ‰ “ (4.52) E e´Zpt1 q pM pt1 q ` ⟨N, M ⟩ pt1 qq G “ E e´Zpt1 q Y pt1 q G for all bounded Ft1 -measurable variables G. So finally we get, P-almost surely, and hence,. e´Zpt1 q Y pt1 q “ e´Zpt1 q pM pt1 q ` ⟨N, M ⟩ pt1 qq ,. Y pt1 q “ M pt1 q ` ⟨N, M ⟩ pt1 q , P-almost surely. This shows assertion (c) and completes the proof of Proposition 4.10.. 257 Download free eBooks at bookboon.com. . Click on the ad to read more.

(22) Advanced stochastic processes: Part II 258 4. STOCHASTIC DIFFERENTIAL EQUATIONS Stochastic differential equations. A combination of Proposition 4.10 and L´evy’s characterization of Brownian motion in Rd yields the following result. 4.12. Proposition. Let the Rd -valued process s ÞÑ cpsq be an adapted process which predictable relative to Brownian motion pBptqqtě0 . Put N ptq “ şt cpsq dBpsq, and 0 żt ż 1 1 t Zptq “ N ptq ` ⟨N, N ⟩ ptq “ cpsq dBpsq ` |cpsq|2 ds, t ě 0. 2 2 0 0 “ ´Zptq ‰ Suppose that for all t ą 0 the equality E e “ 1 holds. Then the process şt pW ptqqtě0 , defined by W ptq “ Bptq ` 0 cpsq ds is Brownian motion relative to the measure A ÞÑ QN pAq, A P FT , as defined in Proposition 4.10. Proof. An application of Proposition 4.10 with M ptq “ Bj ptq shows that the process ⟨ż ⟩ żt p¨q cpsq dBpsq, Bj ptq Wj ptq “ Bj ptq ` cj psq ds “ Bj ptq ` 0. 0. “ Bj ptq ` ⟨N, Bj ⟩ ptq. is a local QN -martingale. Moreover, ⟨Wj1 , Wj2 ⟩ ptq “ δj1 ,j2 t. From Theorem 4.5 we see that the process t ÞÑ W ptq is a QN -Brownian motion. This completes the proof of Proposition 4.12. It will be very convenient to introduce Hermite polynomials phk pxqqkPN , and to establish some of their properties. In the context of stochastic calculus they also play a central role. The Hermite polynomial hk pxq is defined by ˆ ˙k ´ ¯ 1 2 d k 12 x2 hk pxq “ p´1q e e´ 2 x . (4.53) dx For k P N, x P R, a ą 0, we write ˆ ˙ x k{2 Hk px, aq “ a hk ? . a Then we have H0 px, aq “ 1, H1 px, aq “ x, H2 px, aq “ x2 ´ a, H3 px, aq “ x3 ´ ax. The Hermite polynomials satisfy the following recurrence relation: and therefore. hk`2 pxq ´ xhk`1 pxq ` pk ` 1qhk pxq “ 0, k ě 0,. Hk`2 px, aq ´ xHk`1 px, aq ` pk ` 1qaHk px, aq “ 0, k ě 0.. (4.54) (4.55). The equality in (4.54) can be proved by induction and the definition of hk in (4.53). From the definition of hk`1 pxq it follows that h1k`1 pxq “ xhk`1 pxq ´ hk`2 pxq, and so, by (4.54) we see h1k`1 pxq “ pk ` 1qhk pxq, k ě 0.. From (4.54) and (4.56) we infer. hk`2 pxq ´ xhk`1 pxq ` h1k`1 pxq “ 0, k ě 0,. 258 Download free eBooks at bookboon.com. (4.56).

(23) Advanced stochastic processes:TO Part II Stochastic differential 1. SOLUTIONS STOCHASTIC DIFFERENTIAL EQUATIONS 259 equations. and hence. hk`1 pxq ´ xhk pxq ` h1k pxq “ 0, k ě 0. (4.57) By diﬀerentiating the equality in (4.57) and again using (4.56) we obtain the following diﬀerential equation: h2k pxq ´ xh1k pxq ` khk pxq “ 0, k ě 0.. (4.58). In the following proposition we collect some of their properties. 4.13. Proposition. For τ, x P R and a ą 0 the following identities are true: 8 ÿ τk τ x´ 12 τ 2 a e Hk px, aq, “ (4.59) k! k“0 e. τ x´ 12 τ 2. 8 8 ÿ ÿ τk τk Hk px, 1q “ hk pxq, “ k! k! k“0 k“0. (4.60). B 1 B2 B Hk`1 px, aq “ pk ` 1qHk px, aq, and Hk px, aq ` Hk px, aq “ 0. 2 Bx 2 Bx Ba (4.61) ´ ¯ r Proof. Let the sequence hk pxq be such that, for all x and τ P C, the kPN equality 8 ÿ τkr τ x´ 12 τ 2 hk pxq “ (4.62) e k! k“0. holds. Then. r hk pxq “. ˆ. B Bτ. ˙k ´. “ p´1qk. “ hk pxq.. e. ˆ. τ x´ 12 τ 2. B Bx. ˙k ´. ¯ˇ ˇ ˇ. τ “0. “e. 1 2 x 2. ¯ˇ 2 1 ˇ e´ 2 pτ ´xq ˇ. ˆ. τ “0. B Bτ. ˙k ´. “ p´1qk. e. ´ 12 pτ ´xq2. ˆ. d dx. ¯ˇ ˇ ˇ. τ “0. ˙k ´. 1 2. e´ 2 x. ¯. (4.63) ? The equality in (4.63) implies the identity in (4.60). By a correct scaling (τ a x replaces τ , and ? replaces x) the equality in (4.59) follows from (4.60) and a the deﬁnition of Hk px, aq. The equalities in (4.61) follow from (4.56) and from (4.58) respectively. Altogether this completes the proof of Proposition 4.13. In the following proposition the process t ÞÑ M ptq, t P r0, T s, is a martingale on the probability space pΩ, F, Pq. Its quadratic variation process is denoted by t ÞÑ ⟨M, M ⟩ ptq, t P r0, T s. 4.14. Proposition. The following identities hold: żt Hk pM psq, ⟨M, M ⟩ psqq Hk`1 pM ptq, ⟨M, M ⟩ ptqq “ dM psq pk ` 1q! k! 0 ż 1dM ps1 q . . . dM psk`1 q . “ 0ăs1 ă...ăsk`1 ăt. 259 Download free eBooks at bookboon.com. (4.64).

(24) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS 260 4. STOCHASTIC. Stochastic differential equations. In addition, the following equalities hold as well: 1 2. eτ M ptq´ 2 τ ⟨M,M ⟩ptq żt 1 2 “ 1 ` τ eτ M psq´ 2 τ ⟨M,M ⟩psq dM psq “1`. 0 ℓ´1 ÿ. τ. k“1. `τ “. ℓ´1 ÿ. ℓ. ż. k. ż. 0ăs1 ă¨¨¨ăsk ăt. 0ăs1 ă¨¨¨ăsℓ ăt. ż. ż. 1dM ps1 q . . . dM psk q 1 2 ⟨M,M ⟩ps1 q. eτ M ps1 q´ 2 τ. dM ps1 q . . . dM psℓ q. (4.65). τk Hk pM ptq, ⟨M, M ⟩ ptqq k! k“0 ż ż 1 2 ℓ `τ eτ M ps1 q´ 2 τ ⟨M,M ⟩ps1 q dM ps1 q . . . dM psℓ q 0ăs1 ă¨¨¨ăsℓ ăt. ℓ ÿ τk Hk pM ptq, ⟨M, M ⟩ ptqq “ k! k“0 ż ż ´ ¯ 1 2 ℓ `τ eτ M ps1 q´ 2 τ ⟨M,M ⟩ps1 q ´ 1 dM ps1 q . . . dM psℓ q . (4.66) 0ăs1 ă¨¨¨ăsℓ ăt. Please notice that in the equalities in (4.64) through (4.66) the order of integration has to be respected: first we integrate with respect dM ps1 q, then with respect to dM ps2 q and so on.. 260 Download free eBooks at bookboon.com. Click on the ad to read more.

(25) Advanced stochastic processes:TO PartSTOCHASTIC II Stochastic differential 1. SOLUTIONS DIFFERENTIAL EQUATIONS 261 equations. Proof. These equalities follow from Itˆo’s formula and the equalities in Proposition 4.13. Itˆo’s lemma is applied to the functions px, aq ÞÑ Hk`1 px, aq, 1 2 and px, aq ÞÑ eτ x´ 2 τ a with x “ M psq, and a “ ⟨M, M ⟩ psq. In particular the equalities in (4.61) are relevant. This completes the proof of Proposition 4.14. In the following proposition we collect some equalities in case we consider an 1 exponential martingale t ÞÑ eM ptq´ 2 ⟨M,M ⟩ptq in case the process t ÞÑ ⟨M, M ⟩ ptq is deterministic. 4.15. Proposition. Let t ÞÑ M ptq, 0 ď t ď T , be a martingale on pΩ, F, Pq with the property that the variation process t ÞÑ ⟨M, M ⟩ ptq, 0 ď t ď T , is deterministic. The the following identities are true: ﬀ «ż ż ż ż E. 0ăs1 ă¨¨¨ăsk1 ăt. dM ps1 q . . . dM psk1 q ¨. 0ăρ1 ă¨¨¨ăρk2 ăt. dM pρ1 q . . . dM pρk2 q. “ E rHk1 pM ptq, ⟨M, M ⟩ ptqq Hk2 pM ptq, ⟨M, M ⟩ ptqqs. p⟨M, M ⟩ ptqqk1 “ δk1 ,k2 , and k1 ! «ˇż ˇ2 ﬀ ż ˇ ˇ 1 E ˇˇ eM ps1 q´ 2 ⟨M,M ⟩ps1 q dM ps1 q . . . dM psℓ qˇˇ. (4.67). 0ăs1 ă¨¨¨ăsℓ ăt. żt. p⟨M, M ⟩ ptq ´ ⟨M, M ⟩ psqqℓ´1 d ⟨M, M ⟩ psq pℓ ´ 1q! 0 ℓ´1 ÿ p⟨M, M ⟩ ptqqj “ e⟨M,M ⟩ptq ´ . j! j“0. “. e⟨M,M ⟩psq. (4.68). Proof. Let the ”ş and s ÞÑ F2 psq be suchıthat ”ş predictable processes ıs ÞÑ F1 psq T T 2 the quantities E 0 |F1 psq| d ⟨M, M ⟩ psq and E 0 |F2 psq|2 d ⟨M, M ⟩ psq are ﬁnite. Then we have ȷ „ż t2 ȷ „ż t 2 ż t2 E F1 psq dM psq ¨ F2 psq dM psq “ E F1 psqF2 psq d ⟨M, M ⟩ psq , t1. t1. t1. (4.69) for 0 ď t1 ă t2 ď T . By repeatedly employing the equality in (4.69) and using the fact that the process s ÞÑ ⟨M, M ⟩ psq is deterministic we infer, for 1 ď k1 ă k2 , and 0 ă t ď T , with ℓ “ k2 ´ k1 , ﬀ «ż ż ż ż E. 0ăs1 ă¨¨¨ăsk1 ăt. “E. «ż. dM ps1 q . . . dM psk1 q ¨. 0ăs1 ă¨¨¨ăsk2 ăt. ż. 0ăρ1 ă¨¨¨ăρk2 ăt. dM pρ1 q . . . dM pρk2 q. dM ps1 q . . . dM psℓ q d ⟨M, M ⟩ psℓ`1 q . . . d ⟨M, M ⟩ psk2 q. ﬀ. 261 Download free eBooks at bookboon.com.

(26) Advanced stochastic processes: Part II 262. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. “E. «ż. 0ăs1 ă¨¨¨ăsℓ ăt. Stochastic differential equations. ﬀ p⟨M, M ⟩ ptq ´ ⟨M, M ⟩ psℓ qqk1 dM ps1 q . . . dM psℓ q “ 0. k1 ! (4.70). ż. If in (4.70) k1 “ k2 , and so ℓ “ 0, then we obtain »ˇ ˇ2 ﬁ ż k1 ˇż ˇ ˇ ˇ ﬂ p⟨M, M ⟩ ptqq – E ˇ . dM ps1 q . . . dM psk1 qˇ “ ˇ 0ăs1 ă¨¨¨ăsk ăt ˇ k1 ! 1. (4.71). The equalities in (4.70) and (4.71) show the equalities in (4.67). The proof of the equalities requires an induction argument. For ℓ “ 1 we have «ˇż ˇ2 ﬀ ˇ t M ps q´ 1 ⟨M,M ⟩ps q ˇ 1 E ˇˇ e 1 2 dM ps1 qˇˇ 0. “. “ “. żt 0. żt 0. żt 0. “ ‰ E e2M psq´⟨M,M ⟩psq d ⟨M, M ⟩ psq ”. E e. 2M psq´ 12 ⟨2M,2M ⟩psq. ı. e⟨M,M ⟩psq d ⟨M, M ⟩ psq. e⟨M,M ⟩psq d ⟨M, M ⟩ psq “ e⟨M,M ⟩ptq ´ 1.. (4.72). The equalities in (4.72) imply those in (4.68) for ℓ “ 1. The second equality follows by partial integration and induction with respect to ℓ. The ﬁrst equality in (4.68) can be obtained by an argument which is very similar to the proof of the equality in (4.67) with k1 “ k2 “ ℓ. The details are left to the reader. This completes the proof of Proposition 4.15.. . 4.16. Corollary. Let the hypotheses and notation be as in Proposition 4.14. Then ż ż 1 2 ℓ (4.73) eτ M ps1 q´ 2 τ ⟨M,M ⟩ps1 q dM ps1 q . . . dM psℓ q “ 0, lim τ ℓÑ8. 0ăs1 ă¨¨¨ăsℓ ăt. P-almost surely. If the limit in (4.73) is in fact an L1 -limit, then the proτ M ptq´ 12 τ 2 ⟨M,M ⟩ptq cess is a martingale. In particular, it then follows that ı ” t ÞÑ e1 2 E eτ M ptq´ 2 τ ⟨M,M ⟩ptq “ 1; compare with the inequality in (4.41) and with Theorem 4.11. If the process t ÞÑ ⟨M, M ⟩ ptq is real-valued and deterministic, then the limit in (4.73) is an L2 -limit, and so also an L1 -limit.. Proof. Equality (4.73) in Corollary 4.16 follows from the equality in (4.59) in Proposition 4.13 with x “ M ptq and a “ ⟨M, M ⟩ ptq together with the equalities in (4.65) and (4.61). The assertion about the L1 -convergence also follows from these arguments. The only topic that requires some is the one about the situation where the process t ÞÑ ⟨M, M ⟩ ptq is deterministic. In this case the terms in the sum in (4.65) are orthogonal in L2 pΩ, FT , Pq, and this sum 1 2 converges in L2 -sense to eτ M ptq´ 2 τ ⟨M,M ⟩ptq . These assertions follow from the. 262 Download free eBooks at bookboon.com.

(27) Advanced stochastic processes:TO PartSTOCHASTIC II Stochastic differential 1. SOLUTIONS DIFFERENTIAL EQUATIONS 263 equations. identities (4.67) and (4.68) in Proposition 4.15. This completes the proof of Corollary 4.16. The previous results, i.e. Proposition 4.14 and şt Corollary 4.16 are applicable if the martingale M ptq is of the form M ptq “ 0 hpsq ¨ dW psq, where t ÞÑ W ptq şt is standard Brownian motion. Then ⟨M, M ⟩ ptq “ 0 |hpsq|2 ds. If s ÞÑ hpsq is deterministic, then in (4.73) we have L2 -convergence. These martingales play a role in the martingale representation theorem: see Theorem 4.21. 1.4. Weak solutions to stochastic diﬀerential equations. In the following theorem the symbols σi,j and bj , 1 ď i, j ď d, stand for real-valued locally bounded Borel measurable functions defined on r0, 8q ˆ Rd . The matrix pai,j ps, xqqdi,j“1 is defined by aj,k ps, xq “. d ÿ. k“1. σi,k ps, xqσj,k ps, xq “ pσps, xqσ ˚ ps, xqqi,j .. ` ˘ For s ě 0, the operator Lpsq is defined on C 2 Rd with values in the space of locally bounded Borel measurable functions: d d ÿ ` ˘ 1 ÿ ai,j ps, xq Di Dj f pxq ` bj ps, xqDj f pxq, f P C 2 Rd . Lpsqf pxq “ 2 i,j“1 j“1 (4.74). Excellent Economics and Business programmes at:. “The perfect start of a successful, international career.” CLICK HERE. to discover why both socially and academically the University of Groningen is one of the best places for a student to be. www.rug.nl/feb/education. 263 Download free eBooks at bookboon.com. Click on the ad to read more.

(28) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 264 4. STOCHASTIC. The following theorem shows the close relationship between weak solutions and solutions to the martingale problem. 4.17. Theorem. Let pΩ, F, Pq be a probability space with a right-continuous filtration pFt qtě0 . Let tXptq “ pX1 ptq, . . . , Xd ptqq : t ě 0u be a d-dimensional continuous adapted process. Then the following assertions are equivalent: ` ˘ (i) For every f P C 2 Rd the process żt (4.75) t ÞÑ f pXptqq ´ f pXp0qq ´ Lpsqf pXpsqq ds 0. is a local martingale. (ii) The processes żt t ÞÑ Mj ptq :“ Xj ptq ´ bj ps, Xpsqq ds, t ě 0, 1 ď j ď d,. (4.76). 0. are local martingales with covariation processes żt t ÞÑ ⟨Mi , Mj ⟩ ptq “ ai,j ps, Xpsqq ds, t ě 0, 1 ď i, j ď d.. (4.77). 0. On an extended probability space pΩ ˆ Ω1 , Ft b Ft1 , P ˆ P1 q there exists a Brownian motion tBptq : t ě 0u starting at 0 such that żt żt Xptq “ Xp0q ` b ps, Xpsqq ds ` σ ps, Xpsqq dBpsq, t ě 0. (4.78). (iii). 0. 0. Here pΩ1 , Ft1 , P1 q is an independent copy of pΩ, Ft , Pq.ş Moreover, the equality in t (4.78) implies that the stochastic integral pω, ω 1 q ÞÑ 0 σ ps, Xpsqq dBpsq pω, ω 1 q is P ˆ P1 -independent of ω 1 . If the matrix σ ps, yq is invertible, then there is no need for this extension. Examples of (Feller) semigroups can be manufactured by taking a continuous function φ : r0, 8q ˆ E Ñ E with the property that φ ps ` t, xq “ φ pt, φ ps, xqq, for all s, t ě 0 and x P E. Then the mappings f ÞÑ P ptqf , with P ptqf pxq “ f pφ pt, xqq deﬁnes a semigroup. It is a Feller semigroup if limxÑ△ φ pt, xq “ △. An explicit example of such a function, which does not provide a Feller-Dynkin x semigroup on C0 pRq is given by φpt, xq “ b (example due to V. 1 ` 12 tx2 Kolokoltsov [72], and [71]). Put upt, xq “ P ptqf pxq “ f pφpt, xqq. Then Bu Bu pt, xq “ ´x3 pt, xq. In fact this (counter-)example shows that solutions Bt Bx to the martingale problem do not necessarily give rise to Feller-Dynkin semigroups. These are semigroups which preserve not only the continuity, but also the fact that functions which tend to zero at △ are mapped to functions with the same property. However, for Feller semigroups we only require that continuous functions with values in r0, 1s are mapped to continuous functions with the same properties. Therefore, it is not needed to include a hypothesis like. 264 Download free eBooks at bookboon.com.

(29) Advanced stochastic processes:TO Part II Stochastic differential 1. SOLUTIONS STOCHASTIC DIFFERENTIAL EQUATIONS 265 equations. (4.79) which reads as follows: for every pτ, s, t, xq P r0, T s3 ˆ E, τ ă s ă t, the following equality holds: Pτ,x rXptq P Es “ Pτ,x rXptq P E, Xpsq P Es .. (4.79). Nadirashvili [99] constructs an elliptic operator in a bounded open domain U Ă Rd with a regular boundary such that the martingale problem is not uniquely solvable. More precisely the result reads as follows. Consider an elliptic d ÿ B2 operator L “ a2j,k , where aj,k “ aj,k are measurable functions on Rd Bx Bx j k j,k“1 such that d ÿ aj,k ξj ξk ď c |ξ|2 , ξ P Rd , c´1 |ξ|2 ď j,k“1. for some ellipticity constant c ě 1. There exists a diﬀusion pXptq, Px q corresponding to the operator L which can be deﬁned as a solution to the martinşt gale problem P rXp0q “ xs` “˘ 1, f pXptqq ´ f pXp0qq ´ 0 f pXpsqq ds is a Px martingale for all f P C 2 Rd . Nadirashvili is interested in non-uniqueness in the above martingale problem and in non-uniqueness of solutions to the Dirichlet problem Lu “ 0 in Ω, the unit ball in Rd , u “ g on BΩ, where Ω Ă Rd is a bounded domain with smooth boundary and g P C 2 pBΩq. In particular, so-called good solutions u to the Dirichlet problem are investigated. A good solution is a function u which is the limit of a subsequence of solutions un , d ÿ B 2 un anj,k “ 0 in Ω, un “ g on BΩ, where n P N, to the equation Ln un “ Bx Bx j k j,k“1 the operators Ln are elliptic with smooth coeﬃcients anj,k and a common ellipticity constant c such that anj,k Ñ aj,k almost everywhere in Ω as n Ñ 8. The main result is the following theorem: There exists an elliptic operator L of the above form deﬁned in the unit ball B1 Ă Rd , d ě 3, and there is a function g P C 2 pBB1 q such that the formulated Dirichlet problem has at least two good solutions. An immediate consequence is non-uniqueness of solutions to the corresponding martingale problem. The following corollary easily follows from Theorem 4.17. It establishes a close relationship between unique weak solutions to stochastic diﬀerential equations and unique solutions to the martingale problem. For the precise notion of “unique weak solutions” see Deﬁnition 4.19 below. This result should also be compared with Proposition 3.43, where the connection with (strong) Markov processes is explained. 4.18. Corollary. ˘ Let the notation and hypotheses be as in Theorem 4.17. Put ` Ω “ C r0, 8q, Rd , and Xptqpωq “ ωptq, t ě 0, ω P Ω. Fix x P E. Then the following assertions are equivalent: (i) There exists a unique probability measure P on F such that the process żt f pXptqq ´ f pXp0qq ´ Lpsqf pXpsqq ds 0. 265 Download free eBooks at bookboon.com.

(30) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 266 4. STOCHASTIC. is a P-martingale for all C 2 -functions f with compact support, and such that P rXp0q “ xs “ 1. (ii) The stochastic integral equation żt żt Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds (4.80) 0. 0. has unique weak solutions.. 4.19. Definition. The equation in (4.80) is said to have unique weak solutions on the interval r0, T s, also called unique distributional solutions, provided that the ﬁnite-dimensional distributions of the process Xptq, ď t ď T , which satisfy (4.80) do not depend on the particular Brownian motion Bptq which occurs in (4.80). This is the case if and only if for any pair of Brownian motions tpBptq : T ě t ě 0q , pΩ, F, Pqu and tpB 1 ptq : T ě t ě 0q , pΩ1 , F, P1 qu. and any pair of adapted processes tXptq : T ě t ě 0u and tX 1 ptq : T ě t ě 0u for which żt żt Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds and 0 0 żt żt X 1 ptq “ x ` σ ps, X 1 psqq dB 1 psq ` b ps, X 1 psqq ds 0. 0. the ﬁnite-dimensional distributions of the process tXptq : T ě t ě 0u relative to P coincide with the ﬁnite-dimensional distributions of tX 1 ptq : T ě t ě 0u relative to P1 .. In the past four years we have drilled. 89,000 km That’s more than twice around the world.. Who are we?. We are the world’s largest oilfield services company1. Working globally—often in remote and challenging locations— we invent, design, engineer, and apply technology to help our customers find and produce oil and gas safely.. Who are we looking for?. Every year, we need thousands of graduates to begin dynamic careers in the following domains: n Engineering, Research and Operations n Geoscience and Petrotechnical n Commercial and Business. What will you be?. careers.slb.com Based on Fortune 500 ranking 2011. Copyright © 2015 Schlumberger. All rights reserved.. 1. 266 Download free eBooks at bookboon.com. Click on the ad to read more.

(31) Advanced stochastic processes:TO Part II Stochastic differential 1. SOLUTIONS STOCHASTIC DIFFERENTIAL EQUATIONS 267 equations. Proof of Theorem 4.17. (i) ùñ (ii) With fj px1 , . . . , xd q “ xj , 1 ď j ď d, assertion (i) implies that the process żt żt Mj ptq “ Xj ptq ´ bj ps, Xpsqq ds “ fj pXptqq ´ Lpsqfj pXpsqq ds (4.81) 0. 0. is a local martingale. We will show that the processes " * żt Mi ptqMj ptq ´ ai,j ps, Xpsqq ds : t ě 0 , 1 ď i, j ď d, 0. are local martingales as well. To this end fix 1 ď i, j ď d, and define the function fi,j : Rd Ñ R by fi,j px1 , . . . , xd q “ xi xj . From (i) it follows that the process " * żt Xi ptqXj ptq ´ pai,j ps, Xpsqq ` bi ps, Xpsqq Xj psq ` bj ps, Xpsqq Xi psqq ds 0. is a local martingale. For brevity we write. αi,j psq “ ai,j ps, Xpsqq , βj psq “ bj ps, Xpsqq , βi psq “ bi ps, Xpsqq , żs żs Mi psq “ Xi psq ´ βi pτ q dτ, Mj psq “ Xi psq ´ βj pτ q dτ, 0 0 żs pβi pτ qXj pτ q ` βj pτ qXi pτ q ` αi,j pτ qq dτ. Mi,j psq “ Xi psqXj psq ´. (4.82). 0. Then the processes Mi and Mi,j are local martingales. Moreover, we have ˆ ˙ˆ ˙ żt żt Mi ptq ` βi psq ds Mj ptq ` βj psq ds “ Xi ptqXj ptq 0 0 żt “ pβi pτ qXj pτ q ` βj pτ qXi pτ q ` αi,j pτ qq dτ ` Mi,j ptq 0 żt “ pβi pτ q pXj pτ q ´ Mj pτ qq ` βj pτ q pXi pτ q ´ Mi pτ qq ` αi,j pτ qq dτ 0 żt ` pβi pτ qMj pτ q ` βj pτ qMi pτ qq dτ ` Mi,j ptq 0 żt żt “ βi pτ q pXj pτ q ´ Mj pτ qq dτ ` βj pτ q pXi pτ q ´ Mi pτ qq dτ 0 0 żt żt ` αi,j pτ q dτ ` pβi pτ qMj pτ q ` βj pτ qMi pτ qq dτ ` Mi,j ptq 0 0 żt żτ żτ żt βi psq ds dτ βj psq ds dτ ` βj pτ q “ βi pτ q 0 0 0 0 żt żt ` αi,j pτ q dτ ` pβi pτ qMj pτ q ` βj pτ qMi pτ qq dτ ` Mi,j ptq 0 ż ż0 ż ż “ βi pτ qβj psq dτ ds ` βi pτ qβj psq dτ ds 0ăsăτ ăt żt. `. 0. αi,j pτ q dτ `. żt 0. 0ăτ ăsăt. pβi pτ qMj pτ q ` βj pτ qMi pτ qq dτ ` Mi,j ptq. 267 Download free eBooks at bookboon.com.

(32) Advanced stochastic processes: Part II 268. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. “. żt 0. żt. Stochastic differential equations. żt. βi pτ q dτ βj psq ds ` αi,j psq ds ` Mi,j ptq 0 0 żt ` pβi psqMj psq ` βj psqMi psqq ds.. (4.83). 0. Consequently, from (4.83) we see żt Mi ptqMj ptq ´ αi,j psq ds 0 żt “ Mi,j ptq ´ pβi psq pMj ptq ´ Mj psqq ` βj psq pMi ptq ´ Mi psqqq ds.. (4.84). 0. It is readily veriﬁed that the processes żt βi psq pMj ptq ´ Mj psqq ds and 0. żt 0. βj psq pMi ptq ´ Mi psqq ds. are local martingales. It follows that the process " * żt Mi ptqMj ptq ´ αi,j psq ds : t ě 0 0. is a local martingale. So that the covariation process ⟨Mi , Mj ⟩ is given by şt ⟨Mi , Mj ⟩ ptq “ 0 αi,j psq ds. (ii) ùñ (iii) This implication follows from an application of Theorem 4.8 with Φi,j ptq “ ai,j pt, Xptqq, and χptq “ σ pt, Xptqq´1 . If the matrix process σ pt, Xptqq is not invertible we proceed as follows. First we choose a Brownian motion pB 1 ptqqtě0 which is independent of pΩ, Ft , Pq and which lives on the probability space pΩ1 , Ft1 , P1 q. The probability spaces pΩ, Ft , Pq and pΩ1 , Ft1 , P1 q are coupled by employing a standard extension of ¯the original probability space pΩ, Ft , Pq. ´ r r t “ Ft b F1 , and r , where Ω r r “ Ω ˆ Ω1 , F This extension is denoted by Ω, Ft , P t 1 1 1 1 1 1 1 r “ P ˆ P . Finally, B r pt, ω, ω q “ B pt, ω q, t ě 0, pω, ω q P Ω ˆ Ω . We have a P martingale M psq, 0 ď s ď t, on pΩ, Ft , Pq with the properties of assertion (ii). We introduce the matrix processes ψrε psq, ε ą 0, ER psq, and EN psq as follows ψrε psq “ σ ˚ ps, Xpsqq pσ ps, Xpsqq σ ˚ ps, Xpsqq ` εIq´1. ER psq “ lim σ ˚ ps, Xpsqq pσ ps, Xpsqq σ ˚ ps, Xpsqq ` εIq´1 σ ps, Xpsqq , and εÓ0. EN psq “ I ´ ER psq. The matrix ER psq can be considered as an orthogonal projection on the range of the matrix σ ˚ ps, Xpsqq σ ps, Xpsqq, and EN ps as an orthogonal projection on its null space. More precisely, ER psqσ ˚ ps, Xpsqq “ σ ˚ ps, Xpsqq , and σ ps, Xpsqq EN psq “ 0. In terms of these processes we deﬁne the following process: żs żs r EN pτ q dB 1 pτ q. Bpsq “ lim ψε pτ q dM pτ q ` εÓ0. 0. 0. 268 Download free eBooks at bookboon.com. (4.85).

(33) Advanced stochastic processes:TO Part II Stochastic differential 1. SOLUTIONS STOCHASTIC DIFFERENTIAL EQUATIONS 269 equations. Next we şwill prove that the process s ÞÑ Bpsq is a Brownian motion, and that s M psq “ 0 σ pτ, Xpτ qq dBpτ q. Put żs żs r Bε psq “ ψε pτ q dM pτ q ` EN pτ q dB 1 pτ q. (4.86) 0. Then we have:. ⟨Bε,j1 , Bε,j2 ⟩ psq “. d ÿ. żs. k1 ,k2 ,ℓ“1 0. ` ` `. d żs ÿ. k“1 0. d żs ÿ. k“1 0. d żs ÿ. k“1 0. 0. ψrε,j1 ,k1 pτ qψrε,j1 ,k1 pτ qσk1 ,ℓ pτ, Xpτ qq σk2 ,ℓ pτ, Xpτ qq dτ. ψrε,j1 ,k1 pτ qEN,j2 ,K1 pτ q d ⟨Mk1 , Bk1 ⟩ pτ q ψrε,j2 ,k1 pτ qEN,j1 ,K1 pτ q d ⟨Mk1 , Bk1 ⟩ pτ q. EN,j1 ,k pτ qEN,j2 ,k pτ q dτ. (the processes M and B 1 are Pr-independent) żs´ ¯ “ ψrε pτ qσ pτ, Xpτ qq σ ˚ pτ, Xpτ qq ψrε˚ pτ q dτ j1 ,j2 0 żs ˚ pEN pτ qEN ` pτ qqj1 ,j2 dτ.. (4.87). 0. American online LIGS University is currently enrolling in the Interactive Online BBA, MBA, MSc, DBA and PhD programs:. ▶▶ enroll by September 30th, 2014 and ▶▶ save up to 16% on the tuition! ▶▶ pay in 10 installments / 2 years ▶▶ Interactive Online education ▶▶ visit www.ligsuniversity.com to find out more!. Note: LIGS University is not accredited by any nationally recognized accrediting agency listed by the US Secretary of Education. More info here.. 269 Download free eBooks at bookboon.com. Click on the ad to read more.

(34) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 270 4. STOCHASTIC. From (4.87) we infer by continuity and the deﬁnition of ER pτ q that. ⟨Bj1 , Bj2 ⟩ psq “ lim ⟨Bε,j1 , Bε,j2 ⟩ psq εÓ0 żs żs ˚ ˚ “ pER pτ qER pτ qqj1 ,j2 dτ ` pEN pτ qEN pτ qqj1 ,j2 dτ 0 0 żs ˚ pER pτ qER˚ pτ q ` EN pτ qEN pτ qqj1 ,j2 dτ “ 0. (the processes ER pτ q and EN pτ q are orthogonal projections such that ER pτ q ` EN pτ q “ I) “ δj1 ,j2 s.. (4.88). From L´evy’s theorem 4.5 it follows that the process s ÞÑ Bpsq, 0 ď s ď t, is a Brownian motion. In order to ﬁnish the proof şs of the implication (ii) ùñ (iii) we still have to prove the equality M psq “ 0 σ pτ, Xpτ qq dBpτ q. For brevity we write σpτ q “ σ pτ, Xpτ qq. Then by deﬁnition and standard calculations with martingales we obtain: żs żs żs r σ pτ q dBε pτ q “ M psq ´ σpτ qψε pτ q dM pτ q ´ σpτ q EN pτ q dB 1 pτ q M psq ´ 0 0 0 żs ` ˘ “ I ´ σpτ qσ ˚ pτ q pσpτ qσ ˚ pτ q ` εIq´1 dM pτ q 0 żs (4.89) “ ε pσpτ qσ ˚ pτ q ` εIq´1 dM pτ q. 0. From (4.89) together with şs the fact that covariation process of the local martingale M psq is given by 0 σpτ qσ ˚ pτ q dτ , it follows that the covariation matrix of the local martingale ż s. M psq ´. 0. σ pτ q dBε pτ q. is given by żs 2 pσpτ qσ ˚ pτ q ` εIq´1 σpτ qσ ˚ pτ q pσpτ qσ ˚ pτ q ` εIq´1 dτ. ε. (4.90). 0. In addition, we have in spectral sense:. ε 0 ď ε2 pσpτ qσ ˚ pτ q ` εIq´1 σpτ qσ ˚ pτ q pσpτ qσ ˚ pτ q ` εIq´1 ď I, 4 2 and thus in L -sense we have ˆ ˙ żs żs 2 σpτ qBε pτ q “ 0. σpτ q dBpτ q “ L - lim M psq ´ M psq ´ εÓ0. 0. (4.91). (4.92). 0. The equality in (4.92) completes the proof of the implication (ii) ÝÑ (iii). (iii) ùñ (i) Let f : Rd Ñ R be a twice continuously diﬀerentiable function. By Itˆo’s lemma we get żt f pXptqq ´ f pXp0qq ´ Lpsqf pXpsqq ds 0. 270 Download free eBooks at bookboon.com.

(35) Advanced stochastic processes:TO Part II Stochastic differential 1. SOLUTIONS STOCHASTIC DIFFERENTIAL EQUATIONS 271 equations. żt. d ż 1 ÿ t “ ∇f pXpsqq ¨ dXpsq ` Di Dj f pXpsqq d ⟨Xi , Xj ⟩ psq 2 i,j“1 0 0 żt ´ Lpsqf pXpsqq ds 0. “. d ÿ. żt. j“1 0. bj ps, Xpsqq Dj f pXpsqq ds. d ż d 1 ÿ ÿ t σi,k ps, Xpsqq σj,k ps, Xpsqq Di Dj f pXpsqq ds ` 2 i,j“1 k“1 0 żt żt ` ∇f pXpsqq σ ps, Xpsqq dBpsq ´ LpsqF pXpsqq ds. “. żt 0. 0. 0. ∇f pXpsqq σ ps, Xpsqq dBpsq.. (4.93). The final expression in (4.93) is a local martingale. Hence (iii) implies (i). . This completes the proof of Theorem 4.8.. 4.20. Remark. The implication (ii) ùñ (i) in Theorem 4.17 can also be proved directly by using Itˆo calculus. Suppose that the local martingales t ÞÑ Mj ptq, 1 ď j ď d, are defined as in assertion (ii) with covariation processes as in (4.77). Let f be a C 2 -function defined on Rd . Then we have: żt f pXptqq ´ f pXp0qq ´ Lpsqf pXpsqq ds 0. d ż 1 ÿ t Di Dj f pXpsqq d ⟨Xi , Xj ⟩ psq “ ∇f pXpsqq dXpsq ` 2 i,j“1 0 0 żt ´ Lpsqf pXpsqq ds 0 żt żt “ ∇f pXpsqq dM psq ` ∇f pXpsqq b ps, Xpsqq ds. żt. 0. 0. żt d ż 1 ÿ t Di Dj f pXpsqq d ⟨Mi , Mj ⟩ psq ´ Lpsqf pXpsqq ds ` 2 i,j“1 0 0 żt żt “ ∇f pXpsqq dM psq ` ∇f pXpsqq b ps, Xpsqq ds 0. d ÿ. 1 ` 2 i,j“1. “. żt 0. żt 0. 0. Di Dj f pXpsqq ai,j ps, Xpsqq ds ´. żt 0. Lpsqf pXpsqq ds. ∇f pXpsqq dM psq.. (4.94). Assertion (i) is a consequence of equality (4.94).. 271 Download free eBooks at bookboon.com.

(36) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 272 4. STOCHASTIC. 2. A martingale representation theorem In this section we formulate and prove the martingale theorem based on an n-dimensional Brownian motion. Proofs are, essentially speaking, taken from [106]. Let pW psqq0ďsă8 be standard Brownian motion in Rn , and let Ft be the σ-ﬁeld generated by pW psqq0ďsďt augmented with the P-null sets. For h P şt ş 2 1 t L8 pr0, T s; Rn q we write Xh ptq :“ e 0 hpsq¨ dW psq´ 2 0 |hpsq| ds . 4.21. Theorem. Let ΨT be the ş subspace of pΩ, FT , Pq spanned by the expoşT hpsq¨dW psq´ 12 0T |hpsq|2 ds n 0 nentials Xh pT q :“ e , h P L8 simple pr0, T s; R q. Then ΨT is 2 dense in the space L pΩ, FT , Pq.. n n In Theorem 4.21 the space L8 simple pr0, T s; R q consists of those R -valued functions h P L8 pr0, T s; Rn q which can be written in the form ˜ ¸ N N N ÿ ÿ ÿ 1ptk´1 ,tk s psq λj “ 1p0,tj s psqλj , 0 ď s ď T, N P N, (4.95) hpsq “ k“1. j“k. j“1. where, for any N P N, 0 “ t0 ă t1 ă ¨ ¨ ¨ ă tN “ T is an arbitrary partition of the interval r0, T s, and where pλj q1ďjďN are arbitrary vectors in Rn . Observe şT ř that, for such functions h, 0 hpsq ¨ dW psq “ N j“1 λj ¨ W ptj q. Also notice that, şT by Itˆo’s lemma, Xh pT q “ 1 ` 0 Xh psqhpsq ¨ dW psq, h P L8 pr0, T s ; Rn q.. .. 272 Download free eBooks at bookboon.com. Click on the ad to read more.

(37) Advanced stochastic processes: Part II Stochastic differential 2. A MARTINGALE REPRESENTATION THEOREM 273 equations. In the ` proof ˘ of Theorem 4.21 the following notation is employed. The symbol C08 RnˆN stands for the vector space of those C 8 -functions φ deﬁned on all real n ˆ N matrices λ with the property that all functions of the form ` ˘m α λ ÞÑ 1 ` }λ}2HS Dj,kj,k φpλq, m P N, 1 ď j ď N, 1 ď k ď n, αj,k P N, α. are bounded. Here Dj,kj,k stands for the derivative of order αj,k relative to the variable λj,k . The symbol }λ}HS stands for the Hilbert-Schmidt norm of the matrix λ; that is }λ}2HS “. n N ÿ ÿ. j“1 k“1. |λj,k |2 , λ “ pλj,k q1ďjďN,1ďkďn .. ˘ ` nˆN ˘ ` 8 C . Functions of the `form λ˘ ÞÑ exp ´ 21 }λ}2HS belong to the space 0 ` nˆN ˘ R 8 nˆN Observe that C0 R , i.e. the constitutes a dense subspace of C0 R space of complex-valued continuous functions which tend to 0 at 8 equipped with the supremum norm. Proof of Theorem 4.21. This statement is true if there exists no g P L pΩ, FT , Pq, which is perpendicular to all XpT q P ΨT . We start by assuming that there is a g P L2 pΩ, FT , Pq such that g is orthogonal to all variables XpT q P ΨT . This orthogonality means that E rXh pT qgs “ 0, for all n h P L8 simple pr0, T s ; R q. Or, what is the same, ż ş ş T 1 T 2 n (4.96) e 0 hpsq¨dW psq´ 2 0 hpsq ds dP “ 0, for all h P L8 simple pr0, T s ; R q. 2. Ω. The equalities in (4.96) are equivalent to ż ş ş T ´ 12 0T hpsq2 ds n e 0 hpsq¨dW psqpwq g dP “ 0, for all h P L8 e simple pr0, T s ; R q, Ω. which amounts to the same as ż ş T n e 0 hpsq¨dW psq g dP “ 0, for all h P L8 simple pr0, T s ; R q,. (4.97). Ω. By taking h as in (4.95), we see that for all λ “ pλ1 , . . . , λN q P pRn qN “ RnˆN and for all pt1 , . . . , třN q P r0, T sN with 0 “ t0 ă t1 ă ¨ ¨ ¨ ă tN “ T , the following ş řN λj ¨W ptj q ş N λj ¨W ptj q j“1 j“1 g dP “ 0. Next, put Gpλq “ Ω e g dP. equality holds: Ω e nˆN , and thus has an analytic The function λ ÞÑ Gpλq is real analytic on R ş řN zj ¨W ptj q nˆN : Gpzq :“ Ω e j“1 g dP for all z P extension to the complex space C ř n nˆN . Here zj ¨ Wj ptq “ k“1 zj,k Wk ptq, zj “ pz1,k , . . . , zn,k q P Cn . Since C Gpλq “ `0 for λ˘ P RnˆN , it follows that Gpzq “ 0 for z P CnˆN . However, for φ P C08 RnˆN , and with ż ż řN φ p py1 , . . . , yN q “ ... e´i j“1 yj ¨xj φ px1 , . . . , xN q dxN . . . dx1 , Rn Rn loooomoooon N times. 273 Download free eBooks at bookboon.com.

(38) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 274 4. STOCHASTIC. where py1 , . . . , yN q P RnˆN , we see that E rφ pW pt1 q , . . . , W ptN qq gs “. ż. Ω. φ pW pt1 q , . . . , W ptN qq g dP. (inverse Fourier transform) ˙ ż ˆ ż řn 1 i j“1 W ptj q¨yj “ e φpyq p dy g dP p2πqn Rn Ω. (Fubini’s theorem) ż ż ř 1 i n j“1 W ptj q¨yj g dPφpyq “ e p dy p2πqn Rn Ω ż 1 p dy “ 0. Gpiyqφpyq “ p2πqn Rn. (4.98). ` nˆN ˘ 8 From `the monotone class theorem, and the fact the space C R is dense 0 ˘ nˆN in C0 R for the uniform topology, it follows that the equality in (4.98), i.e. the equality (4.99) E rφ pW pt1 q , . . . , W ptN qq gs “ 0 ` ˘ can only be true for all φ P C08 RnˆN , and for all pt1 , . . . , tN q P p0, 8qN , 0 ă t1 ă ¨ ¨ ¨ ă tN “ T , for all N P N, provided that E rF gs “ 0 for all bounded FT -measurable random variables F . Consequently, E rXh pT qgs “ 0 for n 2 all h P L8 simple pr0, T s ; R q if and only if the random variable g P L pΩ, FT , Pq is identically 0. This completes the proof of Theorem 4.21. The following theorem is known as the Itˆo representation theorem. 4.22. Theorem. If the random variable XpT q belongs to L2 pΩ, FT , Pq, then there exists a unique predictable Rn -valued process t ÞÑ F ptq, 0 ď t ď T , for ‰ şT “ which 0 E |F psq|2 ds ă 8 and which is such that żT F psq ¨ dW psq. (4.100) XpT q “ E rXpT qs ` 0. In other words the space "ż T * żT “ ‰ 2 C` F psq ¨ dW psq : s ÞÑ F psq predictable and E |F psq| ds ă 8 0. 0. coincides with L pΩ, FT , Pq. 2. Proof of Theorem ´ 4.22. Let ¯ XpT q be as in ¯ 4.22. Then there ´ Theorem Nk Nk exists double sequences pαj,k qj“1 in C and phj,k qj“1 of elements in kPN kPN ř Nk n L8 simple pr0, T s ; R q such that, with Fk ptq “ j“1 αj,k Xhj,k ptqhj,k ptq and with Xk pT q :“. Nk ÿ. j“1. αj,k Xhj,k pT q “. Nk ÿ. j“1. şT. αj,k e 0. hj,k psq¨dW psq´ 21. şT. 274 Download free eBooks at bookboon.com. 0. 2. |hj,k psq|. ds.

(39) Advanced stochastic processes: Part II. Stochastic differential equations. 2. A MARTINGALE REPRESENTATION THEOREM. “. Nk ÿ. j“1. αj,k `. żT ÿ Nk. 0 j“1. “ E rXk pT qs ` we have. żT 0. αj,k Xhj,k ptqhj,k ptq ¨ dW ptq Fk ptq ¨ dW ptq,. (4.101). “ “ ‰ ‰ 0 “ lim E |XpT q ´ Xk pT q|2 “ lim E |Xℓ pT q ´ Xk pT q|2 kÑ8 k,ℓÑ8 " * żT “ ‰ 2 2 E |Fℓ ptq ´ Fk ptq| dt . “ lim |E rXℓ pT q ´ Xk pT qs| ` k,ℓÑ8. 275. (4.102). 0. From (4.102) we infer that E rXpT qs “ limkÑ8 E rXk pT qs and that there exists a predictable process t ÞÑ F ptq such that żT “ ‰ E |F ptq ´ Fk ptq|2 “ 0. lim (4.103) kÑ8. 0. From (4.103) we obtain 2. L - lim. żT. kÑ8 0. Fk ptq ¨ dW ptq “. żT 0. F ptq ¨ dW ptq.. (4.104). A combination of (4.101), (4.102) and (4.104) yields the equality in (4.100). In ‰ şT “ addition, we have 0 E |F psq|2 ds ă 8, and so the existence part in Theorem 4.22 has been established now. The uniqueness part follows from the Itˆo isometry «ˇż ˇ2 ﬀ ȷ „ż T T ˇ ˇ |F2 psq ´ F1 psq|2 ds “ E ˇˇ pF2 psq ´ F1 psqq ¨ dW psqˇˇ “ 0, E 0. 0. şT. if XpT q ´ E rXpT qs “ 0 F1 psq ¨ dW psq “ completes the proof of Theorem 4.22.. şT 0. F2 psq ¨ dW psq. Altogether this . Next we formulate and prove the martingale representation theorem. 4.23. Theorem. Let pM ptqq0ďtďT belong to M2 pΩ, FT , Pq. Then there exists a unique predictable Rn -valued process t ÞÑ ζptq “ pζ1 ptq, . . . , ζn ptqq, ζptq P L2 pΩ, Ft , P; Rn q, such that żt n żt ÿ M ptq “ M p0q ` ζpsq ¨ dW psq “ M p0q ` ζj psq dWj psq. (4.105) 0. j“1 0. Of course, if in Theorem 4.23 the process t ÞÑ M ptq, 0 ď t ď T , is a martingale vector inşRd , then we obtain a predictable matrix Zptq P Rnˆd such that M ptq “ M p0q ` Zpsq˚ dW psq. (This is what one needs in the context of Backward Stochastic Diﬀerential Equations or BSDEs for short.) Proof of Theorem 4.23. Let pM ptqq0ďtďT be as in Theorem 4.23. Theorem 4.22 yields the existence of a unique predictable Rn -valued process t ÞÑ. 275 Download free eBooks at bookboon.com.

(40) Advanced stochastic processes: Part II 276. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. Stochastic differential equations. ζptq “ pζ1 ptq, . . . , ζn ptqq, ζptq P L2 pΩ, Ft , P; Rn q, such that żT n żT ÿ ζpsq¨dW psq “ E rM pT qs` ζj psq dWj psq. (4.106) M pT q “ E rM pT qs` 0. j“1 0. şt. Since the processes t ÞÑ M ptq and t ÞÑ 0 ζj psq dWj psq, 1 ď j ď n, 0 ď t ď T , are martingales, from (4.106) we infer ȷ „ żT ˇ ‰ ˇ “ ˇ ˇ M ptq “ E M pT q Ft “ E E rM pT qs ` ζpsq ¨ dW psq Ft 0 żt żt “ E rM pT qs ` ζpsq ¨ dW psq “ E rM p0qs ` ζpsq ¨ dW psq (4.107) 0. 0. From (4.107) we get M p0q “ E rM p0qs, and so the representation in (4.105) follows from (4.107). The proof of Theorem 4.23 is complete now. . Join the best at the Maastricht University School of Business and Economics!. Top master’s programmes • 3 3rd place Financial Times worldwide ranking: MSc International Business • 1st place: MSc International Business • 1st place: MSc Financial Economics • 2nd place: MSc Management of Learning • 2nd place: MSc Economics • 2nd place: MSc Econometrics and Operations Research • 2nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier ‘Beste Studies’ ranking 2012; Financial Times Global Masters in Management ranking 2012. Maastricht University is the best specialist university in the Netherlands (Elsevier). Visit us and find out why we are the best! Master’s Open Day: 22 February 2014. www.mastersopenday.nl. 276 Download free eBooks at bookboon.com. Click on the ad to read more.

(41) Advanced stochastic processes: II 3. Part GIRSANOV TRANSFORMATION. Stochastic differential 277 equations. 3. Girsanov transformation In this section we want to discuss the Cameron-Martin-Girsanov transformation or just Girsanov transformation. Let pΩ, F, Pq be a probability space with a ﬁltration pFt qtě0 . In addition, let the process tBptq : t ě 0u be a d-dimensional Brownian motion. Let bj , cj , σi,j be Borel measurable locally bounded functions on Rd . Suppose that the stochastic diﬀerential equation żt żt (4.108) Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds 0. 0. has unique weak solutions. For a precise deﬁnition of the notion of “unique weak solutions” see Deﬁnition 4.19. For more information on transformations ¨ unel and Zakai [139]. In particular of measures on Wiener space see e.g. Ust¨ these observations mean that if in equation (4.109) below (for the process Y ptq) the process B 1 ptq is a Brownian motion relative to a probability measure P1 , then the P1 -distribution of the process Y ptq coincides with the P-distribution of the process Xptq which satisﬁes (4.108). Next we will elaborate on this item. Suppose that the process t ÞÑ Y ptq satisﬁes the equation: żt żt Y ptq “ x ` σ ps, Y psqq dBpsq ` pb ps, Y psqq ` σ ps, Y psqq c ps, Y psqqq ds 0 0 żt żt (4.109) “ x ` σ ps, Y psqq dB 1 psq ` b ps, Y psqq ds, 0. 1. şt. 0. where B ptq “ Bptq ` 0 c ps, Y psqq ds. The following proposition says that relative to a martingale transformation P1 of the measure P (Girsanov or CameronMartin transformation) the process t ÞÑ B 1 ptq is a P1 -Brownian motion. More precisely, we introduce the local martingale M 1 ptq and the corresponding martingale measure P1 by ˙ ˆ żt ż 1 t 2 1 |c ps, Y psqq| ds and (4.110) M ptq “ exp ´ c ps, Y psqq dBpsq ´ 2 0 0 P1 rAs “ E rM 1 ptq1A s , A P Ft . (4.111) We also need the process Z 1 ptq deﬁned by ż żt 1 t 1 |c ps, Y psqq|2 ds. Z ptq “ ´ c ps, Y psqq dBpsq ´ 2 0 0. (4.112). In addition, we have a need for a vector function c1 pt, yq satisfying cpt, yq “ c1 pt, yqσpt, yq. We assume that such a vector function c1 pt, yq exists.. 4.24. Proposition. Suppose that the process Y ptq satisfies the equation in (4.109). Let the processes M 1 ptq and Z 1 ptq be defined by (4.110) and (4.112) respectively. Then the following assertions are true: (1) The process t ÞÑ M 1 ptq is a local P-martingale. It is a martingale provided E rM 1 ptqs “ 1 for all t ě 0. (2) Fix t ą 0. The variable M 1 ptq only depends on the process s ÞÑ Y psq, 0 ď s ď t.. 277 Download free eBooks at bookboon.com.

(42) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 278 4. STOCHASTIC. (3) Suppose that the process t ÞÑ M 1 ptq is a P-martingale, and not just a local P-martingale. Then P1 can be considered as a probability measure on the σ-field generated by Ytą0 Ft . (4) Suppose that the process t ÞÑ M 1 ptq is a P-martingale. Then the process t ÞÑ B 1 ptq is a Brownian motion relative to P1 . Proof. 1 From Itˆo calculus we get żt 1 1 M ptq ´ M p0q “ ´ M 1 psqc ps, Y psqq dBpsq, 0. and hence assertion 1 follows, because stochastic integrals with respect to Brownian motion are local martingales. Next we choose a sequence of stopping times τn which increase to 8 P-almost surely, and which are such that the processes t ÞÑ M 1 pt ^ τn q are genuine martingales. Then we see E rM 1 pt ^ τn qs “ 1 for all n P N and t ě 0. Fix t2 ą t1 . Since the processes t ÞÑ M 1 pt ^ τn q, n P N, are P-martingales, we see that ˇ ‰ “ (4.113) E M 1 pt2 ^ τn q ˇ Ft1 “ M 1 pt1 ^ τn q P-almost surely. In (4.113) we let n Ñ 8, and apply Scheﬀ´e’s theorem to conclude that ˇ “ ‰ E M 1 pt2 q ˇ Ft1 “ M 1 pt1 q P-almost surely. (4.114). The equality in (4.114) shows that the process t ÞÑ M 1 ptq is a P-martingale provided that E rM 1 ptqs “ 1 for all t ě 0. This completes the proof of assertion 1. 2 This assertion follows from the following calculation: ż żt 1 t 1 |c ps, Y psqq|2 ds Z ptq “ ´ c ps, Y psqq dBpsq ´ 2 0 0 żt żt 1 “ ´ c ps, Y psqq dB 1 psq ` |c ps, Y psqq|2 ds 2 0 0 (cps, yq “ c1 ps, yqσps, yq) ż żt 1 t 1 |c ps, Y psqq|2 ds “ ´ c1 ps, Y psqq σ ps, Y psqq dB psq ` 2 0 0 ˆż s ˙ ż żt 1 t 1 σ pτ, Y pτ qq dB pτ q ` |c ps, Y psqq|2 ds “ ´ c1 ps, Y psqq d 2 0 0 0 ˆ ˙ żt ż żs 1 t “ ´ c1 ps, Y psqq d Y psq ´ b pτ, Y pτ qq dτ ` |c ps, Y psqq|2 ds. 2 0 0 0 (4.115) From (4.115), (4.110), and (4.112) it is plain that M 1 ptq only depends on the path tY psq : 0 ď s ď tu. 3 This assertion is a consequence of Kolmogorov’s extension theorem. The measure is P1 is well deﬁned on Ytą0 Ft . Here we use the martingale property. By Kolmogorov’s extension theorem, it extends to the σ-ﬁeld generated by this union.. 278 Download free eBooks at bookboon.com.

(43) Advanced stochastic processes: Part II 3. GIRSANOV TRANSFORMATION. Stochastic differential 279 equations. şt 4 The equality B 1 ptq “ Bptq ` 0 c ps, Y psqq ds entails the following equality for the quadratic covariation of the processes Bi1 and Bj1 : ⟨ 1 1⟩ Bi , Bj ptq “ ⟨Bi , Bj ⟩ ptq “ tδi,j . (4.116) From Itˆo calculus we also infer. M 1 ptqBi1 ptq żt żt 1 1 1 “ M psqBi psq dZ psq ` M 1 psq dBi1 psq 0 0 żt żt 1 1 1 1 1 ` M psqB psq d ⟨Z , Z ⟩ psq ` M 1 psqd ⟨Z 1 , Bi1 ⟩ psq 2 0 0 żt ż 1 t 1 1 1 “ ´ M psqBi psqc ps, Y psqq dBpsq ´ M psqBi1 psq |c ps, Y psqq|2 ds 2 0 0 żt żt 1 2 1 1 M psqBi psq |c ps, Y psqq| ds ` M 1 psq dBi psq ` 2 0 0 żt żt ` M 1 psqci ps, Y psqq ds ´ M 1 psqci ps, Y psqq ds 0 0 żt żt 1 1 “ ´ M psqBi psqc ps, Y psqq dBpsq ` M 1 psq dBi psq. (4.117) 0. 0. Upon invoking Theorem 4.5 and employing (4.116) and (4.117) assertion 4 follows. . This concludes the proof of Proposition 4.24.. > Apply now redefine your future. - © Photononstop. AxA globAl grAduAte progrAm 2015. axa_ad_grad_prog_170x115.indd 1. 19/12/13 16:36. 279 Download free eBooks at bookboon.com. Click on the ad to read more.

(44) Advanced stochastic processes: Part II 280 4. STOCHASTIC DIFFERENTIAL EQUATIONS Stochastic differential equations. Let the process Xptq solve the equation in (4.108), and put ˆż t ˙ ż 1 t 2 M ptq “ exp c ps, Xpsqq dBpsq ´ |c ps, Xpsqq| ds , 2 0 0. (4.118). and assume that the process M ptq is not merely a local martingale, but a genuine P-martingale. 4.25. Theorem. Fix T ą 0, and let the functions. bps, yq, σps, yq, cps, yq, and c1 ps, yq, 0 ď s ď T,. be locally bounded Borel measurable vector or matrix functions such that cps, yq “ c1 ps, yqσ ps, yq, 0 ď s ď T , y P Rd . Suppose that the equation in (4.108) possesses unique weak solutions on the interval r0, T s. Uniqueness. If weak solutions to the stochastic diﬀerential equation in (4.109) exist, then they are unique in the sense as explained next. In fact, let the couple pY psq, Bpsqq, 0 ď s ď t, be a solution to the equation in (4.109) with the property that the local martingale M 1 ptq given by ˆ żt ˙ ż 1 t 2 1 M ptq “ exp ´ c ps, Y psqq dBpsq ´ |c ps, Y psqq| ds . (4.119) 2 0 0 satisﬁes E rM 1 ptqs “ 1. Then the ﬁnite-dimensional distributions of the process Y psq, 0 ď s ď t, are given by the Girsanov or Cameron-Martin transform: E rf pY pt1 q , . . . , Y ptn qqs “ E rM ptqf pX pt1 q , . . . , X ptn qqs ,. (4.120). t ě tn ą ¨ ¨ ¨ ą t1 ě 0, where f : Rd ˆ ¨ ¨ ¨ ˆ Rd Ñ R is an arbitrary bounded Borel measurable function. Existence. Conversely, let the process s ÞÑ pXpsq, Bpsqq be a solution to the equation in (4.108). Suppose that the local martingale s ÞÑ M psq, deﬁned by ˙ ˆż s ż 1 s 2 c pτ, Xpτ qq dBpτ q ´ |c pτ, Xpτ qq| dτ , 0 ď s ď t, M psq “ exp 2 0 0 (4.121) ´ ¯ r r is a martingale, i.e. E rM ptqs “ 1. Then there exists a couple Y psq, Bpsq , r 0 ď s ´ď t, where ¯ s ÞÑ Bpsq, 0 ď s ď t, is a Brownian motion on a probability r P r such that r F, space Ω, żs ´ żs ´ ¯ ¯ ´ ¯ r r r r σ τ, Y pτ q dBpτ q ` σ τ, Y pτ q c τ, Y pτ q dτ 0 0 żs ´ ¯ r ` b τ, Y pτ q dτ, (4.122). Yr psq “ x `. 0. and such that „ ˆ żt ´ żtˇ ´ ¯ ¯ˇ2 ˙ȷ 1 ˇ ˇ r exp ´ c s, Yr psq dBpsq r ´ “ 1. E ˇc s, Yr psq ˇ ds 2 0 0 280 Download free eBooks at bookboon.com. (4.123).

(45) Advanced stochastic processes: II 3. Part GIRSANOV TRANSFORMATION. Stochastic differential 281 equations. 4.26. Remark. The formula in (4.120) is known as the Girsanov transform or Cameron-Martin transform of the measure P. It is a martingale measure. Suppose that the process t ÞÑ M 1 ptq, as defined in (4.110) is a P-martingale. Then the proof of Theorem 4.25 shows that the process t ÞÑ M ptq, as defined in (4.118) is a P-martingale. By assertion 1 in Proposition 4.24 the process t ÞÑ M 1 ptq is a P-martingale if and only E rM 1 ptqs “ 1 for all T ě t ě 0, and a similar statement holds for the process t ÞÑ M ptq. If the process t ÞÑ M 1 ptq is a martingale, then taking G ” 1 in (4.135) shows that E rM ptqs “ 1, and hence by 1 in Proposition 4.24 the process t ÞÑ M ptq is a P-martingale. Conversely, if the process t ÞÑ M ptq is a P-martingale, then we reverse the implications in the proof of Theorem 4.25 and take F ” 1 in (4.139) to conclude that E rM 1 ptqs “ 1 for all ě 0. But then the process t ÞÑ M 1 ptq is a P-martingale. Notice that the process t ÞÑ„ M ptq Novikov’s conˆ isż ta P-martingale provided ˙ȷ 1 dition is satisfied, i.e. if E exp |c ps, Xpsqq|2 ds ă 8. For a precise 2 0 formulation see Corollary 4.27 below. Define 1. EpM qptq “ eM ptq´ 2 xM,M yptq . ˙ȷ „ ˆ 1 ⟨M, M ⟩ ptq ă 8, then 4.27. Corollary. If sup E exp 2 tě0 „ ˆ ˙ȷ 1 E exp M p8q ´ ⟨M, M ⟩ p8q “ 1, 2. (4.124). and consequently the process t ÞÑ EpM qptq is a P-martingale relative to the filtration pFt qtě0 , where Ft “ σ pM psq : 0 ď s ď tq, the σ-field generated by the variables M psq, 0 ď s ď t. Novikov’s result is a consequence of results in [76]; see Chapter 1 of [146]. Observe that M p8q “ limtÑ8 M ptq exists P-almost surely. 4.28. Remark. Let s ÞÑ cpsq be a process which is adapted to Brownian d ρ ą 0 be such that Novikov’s condition is satisfied: motion ” ´in Rş , and let¯ı 2 1 2 t E exp 2 ρ 0 |cpsq| ds ă 8. From assertion 4 in Proposition 4.24 and Theorem 4.25 we see that the following ` d ˘nidentity holds for all bounded Borel measurable functions F defined on R : E rF pYρ pt1 q , . . . , Yρ ptn qqs „ ˆ żt ˙ ȷ ż 1 2 t 2 “ E exp ρ cpsqdBpsq ´ ρ |cpsq| ds F pB pt1 q , . . . , B ptn qq 2 0 0 (4.125) şτ where 0 ď t1 ă ¨ ¨ ¨ ă tn ď t, and Yρ pτ q “ Bpτ q ` ρ 0 cpsq ds, 0 ď τ ď t. In particular, if n “ 1 we get ˙ȷ „ ˆ żt E F Bptq ` ρ cpsq ds 0. 281 Download free eBooks at bookboon.com.

(46) Advanced stochastic processes: Part II 282. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. Stochastic differential equations. „. ˆ żt ˙ ȷ ż 1 2 t 2 “ E exp ρ cpsqdBpsq ´ ρ |cpsq| ds F pB ptqq (4.126) 2 0 0 Assume that the gradient DF of the function F exists and is bounded. The equality in (4.126) can be diﬀerentiated with respect to ρ to obtain: „⟨ ˆ ˙ żt ⟩ȷ żt E DF Bptq ` ρ cpsq ds , cpsq ds 0 0 ˙ „ ˆ żt ż 1 2 t 2 |cpsq| ds “ E exp ρ cpsq dBpsq ´ ρ 2 0 0 ˙ ȷ ˆż t żt 2 cpsq dBpsq ´ ρ |cpsq| ds F pBptqq . (4.127) ˆ 0. 0. The bracket in the left-hand side of (4.127) indicates the inner-product in Rd . In (4.127) we put ρ “ 0 and we obtain the ﬁrst order version of the famous integration by parts formula: „ż t ȷ „⟨ ⟩ȷ żt “E cpsq dBpsq F pBptqq . (4.128) E DF pBptqq , cpsq ds 0. 0. We mention that the Cameron-Martin-Girsanov transformation is a cornerstone for the integration by parts formula, which is a central issue in Malliavin calculus. For details on this subject see e.g. Nualart [103, 102], Malliavin [92], Sanz-Sol´e [118], Kusuoka and Stroock [78, 79, 80], Stroock [127], and Norris [100].. 282 Download free eBooks at bookboon.com. Click on the ad to read more.

(47) Advanced stochastic processes: II 3. Part GIRSANOV TRANSFORMATION. Stochastic differential 283 equations. For a proof of Theorem 4.25 we will need the Skorohod-Dudley-Wichura representation ˘ see Theorem 11.7.2 of Dudley [41]. It will be applied with ` theorem: d S “ C r0, ts, R and can be formulated as follows.. 4.29. Theorem. Let pS, dq be a complete separable metric space (i.e. a Polish space), and let Pk , k P N, and P be probability measures on the ş BS of ş Borel field S such that the weak limit w´ limkÑ8 Pk “ P, i.e. limkÑ8 F dPk “ F dP for all bounded continuous functions of F P Cb pSq. Then there exist a probability ´ ¯ r r r space Ω, F, P and S-valued stochastic variables Yrk , k P N, and Yr , defined on r with the following properties: Ω ” ı ” ı r Yrk P B , k P N, and P rBs “ P r Yr P B , B P BS . (1) Pk rBs “ P r surely. (2) The sequence Yrk , k P N, converges to Yr P-almost. 4.30. Remark. An analysis of the existence part of the proof of Theorem 4.25 r psq, shows that the invertibility of the matrix σ ps, yq is not needed. ´Let N ¯ r r r 0 ď s ď t, be a local martingale on a ﬁltered probability space Ω, Fs , P , ´ ¯ r s is generated by Yr pτ q : 0 ď τ ď s . Suppose that the where the σ-ﬁeld F r psq is given by covariation process of N ż ¯ ´ ¯¯ ⟩ ⟨ s´ ´ rj2 psq “ rj1 , N σ τ, Yr pτ q σ ˚ τ, Yr pτ q dτ, 1 ď j1 , j2 ď d. N 0. ´. j1 ,j2. ¯. r P r . Then by assertion (iii) in Theorem r F, Here Yr is a local martingale on Ω, r 4.17 there exists a Brownian motion Bpsq, 0 ď s ď t, on this space such that ż żs ´ ¯ ´ ¯ ´ ¯ s r pτ q “ r q c1 τ, Yr pτ q dN c1 τ, Yr pτ q σ τ, Yr pτ q dBpτ 0 ż0s ´ ¯ r q. c τ, Yr pτ q dBpτ (4.129) “ 0. Proof of Theorem 4.25. Uniqueness. Let the process Y psq, 0 ď s ď t, be a solution to equation (4.109). So that żs żs σ pτ, Y pτ qq dBpτ q ` pb pτ, Y pτ qq ` σ pτ, Y pτ qq c pτ, Y pτ qqq dτ Y psq “ x ` 0 0 żs żs 1 “x` σ pτ, Y pτ qq dB pτ q ` b pτ, Y pτ qq dτ. (4.130) `. 0. ˘. 0. Let F pY psqq0ďsďt be a bounded stochastic variable which depends on the path Y psq, 0 ď s ď t. As observed in 4 of Proposition 4.24 the process B 1 ptq is a P1 -Brownian motion, provided E rM 1 ptqs “ 1. Uniqueness of weak solutions to equation (4.108) implies that the P 1 -distribution of the process s ÞÑ Y psq, 0 ď s ď t, coincides with the P-distribution of the process s ÞÑ Xpsq, 0 ď s ď t. In other words we have “ ` ˘‰ E1 F pY psqq0ďsďt 283 Download free eBooks at bookboon.com.

(48) Advanced stochastic processes: Part II 284. 4. STOCHASTIC DIFFERENTIAL EQUATIONS. Stochastic differential equations. „. ˆ żt ˙ ȷ ż ` ˘ 1 t 2 Y “ E exp ´ c1 ps, Y psqq dN psq ´ |c ps, Y psqq| ds F pY psqq0ďsďt 2 0 0 “ ` ˘‰ “ E F pXpsqq0ďsďt , (4.131). where. żs. Y. σ pτ, Y pτ qq c pτ, Y pτ qq dτ ´ N psq “ Y psq ´ 0 żs σ pτ, Y pτ qq dBpτ q. “. żs 0. b pτ, Y pτ qq dτ (4.132). 0. With ˘ ` G pY psqq0ďsďt ˆ żt ˙ ż ` ˘ 1 t 2 Y |c ps, Y psqq| ds F pY psqq0ďsďt “ exp ´ c1 ps, Y psqq dN psq ´ 2 0 0. we have ` ˘ F pY psqq0ďsďt ˆż t ˙ żt ` ˘ 1 2 “ exp |c ps, Y psqq| ds G pY psqq0ďsďt c1 ps, Y psqq dN Y psq ` 0 0 2. So, since. dN X psq “ dXpsq ´ σ ps, Xpsqq c ps, Xpsqq ds ´ b ps, Xpsqq ds “ σ ps, Xpsqq pdBpsq ´ c ps, Xpsqq dsq. (4.133). it follows that ˘ ` F pXpsqq0ďsďt ˆż t ˙ ż ` ˘ 1 t 2 X “ exp c1 ps, Xpsqq dN psq ` |c ps, Xpsqq| ds G pXpsqq0ďsďt 2 0 0 ˙ ˆż t żt ` ˘ 1 2 c ps, Xpsqq dBpsq ´ |c ps, Xpsqq| ds G pXpsqq0ďsďt . “ exp 2 0 0 (4.134) From (4.131) and (4.134) we infer: “ ` ˘‰ E G pY psqq0ďsďt (4.135) ȷ ˙ „ ˆż t żt ` ˘ 1 2 c ps, Xpsqq ds ´ |c ps, Xpsqq| ds G pXpsqq0ďsďt . “ E exp 2 0 0 By inserting G ” 1 in (4.135) we see that „ ˆż t ˙ȷ ż 1 t 2 E exp c ps, Xpsqq ds ´ |c ps, Xpsqq| ds “1 2 0 0. in case there is a unique solution to the equation in (4.122). This proves the uniqueness part of Theorem 4.25.. 284 Download free eBooks at bookboon.com.

(49) Advanced stochastic processes: II 3. Part GIRSANOV TRANSFORMATION. Stochastic differential 285 equations. Existence. Therefore we will approximate the solution Y by a sequence Yk , k P N, which are solutions to equations of the form: żs σ pτ, Yk pτ qq dBpτ q Yk psq “ x ` 0 żs ` pb pτ, Yk pτ qq ` σ pτ, Yk pτ qq ck pτ, Yk pτ qqq dτ 0 żs żs 1 σ pτ, Yk pτ qq dBk pτ q ` b pτ, Yk pτ qq dτ. (4.136) “x` 0. 0. Here. Bk1 psq. “ Bk psq `. żs 0. ck pτ, Yk pτ qq dτ,. and the coeﬃcients ck ps, yq “ c1,k ps, yqσ ps, yq are chosen in such a way that they are bounded and that cps, yq “ limkÑ8 ck ps, yq for all s P r0, ts and y P Rd . By Novikov’s theorem the corresponding local martingales Mk1 , given by ˆ żs ˙ ż 1 s 2 1 Mk psq “ exp ´ ck pτ, Yk pτ qq dBpτ q ´ |ck pτ, Yk pτ qq| dτ , k P N, 2 0 0. are then automatically genuine martingales: see Corollary 4.27. From the uniqueness of weak solutions to equations in Xptq of the form (4.108) (and thus to equations in Yk psq of the form (4.136) we infer “ ` ˘‰ “ ` ˘‰ E1k F pYk psqq0ďsďt “ E F pXpsqq0ďsďt . (4.137). In equality (4.137) the process Yk psq, 0 ď s ď t, solves the equation in (4.136). The equality in (4.137) can be rewritten as “ ` ˘‰ “ ` ˘‰ E Mk1 ptqF pYk psqq0ďsďt “ E F pXpsqq0ďsďt . (4.138). By (4.115) the equality in (4.138) can be rewritten as ȷ ˙ „ ˆ żt ż ` ˘ 1 t 2 |ck ps, Yk psqq| ds F pYk psqq0ďsďt E exp ´ ck ps, Yk psqq dBpsq ´ 2 0 0 ˆ ˙ „ ˆ żt żs b pτ, Yk pτ qq dτ “ E exp ´ c1,k ps, Yk psqq d Yk psq ´ 0 0 ȷ ˙ ż ` ˘ 1 t 2 |ck ps, Yk psqq| ds F pYk psqq0ďsďt ` 2 0 “ ` ˘‰ “ E F pXpsqq0ďsďt . (4.139) ˘ ` Let G pYk psqq0ďsďt be a (bounded) stochastic variable which depends on the path Yk psq, 0 ď s ď t. From the equality in (4.139) we infer “ ` ˘‰ E G pYk psqq0ďsďt „ ˆż t ˙ ˆ żs “ E exp c1,k ps, Xpsqq d Xpsq ´ b pτ, Xpτ qq dτ 0 0 ȷ ˙ ż ` ˘ 1 t 2 |ck ps, Xpsqq| ds G pXpsqq0ďsďt ´ 2 0 285 Download free eBooks at bookboon.com.

(50) Advanced stochastic processes: Part II 286 4. STOCHASTIC DIFFERENTIAL EQUATIONS Stochastic differential equations. „. ˆż t. 1 “ E exp c1,k ps, Xpsqq σ ps, Xpsqq dBpsq ´ 2 0 ȷ ` ˘ ˆ G pXpsqq0ďsďt “ ` ˘‰ “ E Mk ptqG pXpsqq0ďsďt .. żt 0. 2. |ck ps, Xpsqq| ds. ˙. (4.140). Here the martingales Mk psq are given by ˆż s ˙ ż 1 s 2 ck pτ, Xpτ qq dBpτ q ´ |ck pτ, Xpτ qq| dτ , k P N, Mk psq “ exp 2 0 0. This fact together with the pointwise convergence of Mk psq to M psq, as k Ñ 8, and invoking the hypothesis “ that E ` rM ptqs “ 1,˘‰shows that the right-hand side of (4.140) converges to E M ptqG pXpsqq0ďsďt . In other words the distribuM,X tion PYk of Yk converges weakly to the measure PM,X deﬁned ` by Pd ˘ pAq “ E rM ptq, X P As, where A is a Borel subset of the space C r0, ts, R . By the Skorohod-Dudley-Wichura representation theorem (Theorem 4.29) there exist ´ ¯ ˘ ` r r r a probability space Ω, F, P and C r0, ts, Rd -valued stochastic variables Yrk , r with the following properties: k P N, and Yr , deﬁned on Ω ” ı ” ı r Yrk P B , k P N, and PM,X rBs “ P r Yr P B , for all (1) PYk rBs “ P B P BCpr0,ts,Rd q´ . ¯ r (2) The sequence Yrk converges to Yr P-almost surely. kPN. Need help with your dissertation? Get in-depth feedback & advice from experts in your topic area. Find out what you can do to improve the quality of your dissertation!. Get Help Now. Go to www.helpmyassignment.co.uk for more info. 286 Download free eBooks at bookboon.com. Click on the ad to read more.

(51) Advanced stochastic processes: II 3. Part GIRSANOV TRANSFORMATION. Stochastic differential 287 equations. By taking the limit in (4.140) for k Ñ 8 and using the theorem of SkorohodDudley-Wichura we obtain ˙ȷ „ ˆ´ ¯ “ ` ˘‰ “ E M ptqG pXpsqq0ďsďt E G Yr psq (4.141) 0ďsďt. ` ˘ where G is a bounded continuous function on C r0, ts, Rd . Then we consider r psq, 0 ď s ď t, deﬁned by the process N żs ´ żs ´ ¯ ´ ¯ ¯ r psq “ Yr psq ´ N σ τ, Yr pτ q c τ, Yr pτ q dτ ´ b τ, Yr pτ q dτ. (4.142) 0. 0. r psq would be N Y psq, given by the formula If Yr psq were Y psq, then by (4.130) N in (4.132). Hence the process s ÞÑ N Y psq, s P r0, ts, is a stochastic integral relative to Brownian motion on the space pΩ, Ft , Pq. We want ´to do same ¯ for r P r . Let r psq, 0 ď s ď t, on the probability space Ω, r F, the process s ÞÑ N. PM ptq be the probability measure on pΩ, Ft q deﬁned by PM ptq rAs “ E rM ptq, As, A P Ft . şThen like in item (4) of Proposition 4.24 we see that the process s ÞÑ s Bpsq ´ 0 σ pτ, Xpτ qq dτ is a PM ptq -Brownian motion. In addition, from (4.141) r r psq, 0 ď s ď t, is and (4.142) we infer that the P-distribution of the process N M ptq given by the P -distribution of the process żs żs σ pτ, Xpτ qq c pτ, X pτ qq dτ ´ b pτ, Xpτ qq dτ s ÞÑXpsq ´ 0 0 żs σ pτ, Xpτ qq pdBpτ q ´ c pτ, Y pτ qq dτ q “ 0 żs “ σ pτ, Xpτ qq dB M ptq pτ q, (4.143) 0. where B M ptq psq is a PM ptq -Brownian motion: see Proposition 4.24 item (4). It also follows that the process in (4.143) has covariation process given by the square matrix process żs σ pτ, Xpτ qq σ ˚ pτ, Xpτ qq dτ, 0 ď s ď t. s ÞÑ 0. r psq, 0 ď s ď t, is a local Pr-martingale with Consequently, the process s ÞÑ N covariation process given by żs ´ ¯ ¯ ´ (4.144) σ τ, Yr pτ q σ ˚ τ, Yr pτ q dτ, 0 ď s ď t. s ÞÑ 0. In order to prove (4.144) we must show that the process d żs ´ ¯ ´ ¯ ÿ r r s ÞÑ Nj1 psqNj2 psq ´ σj1 ,k τ, Yr pτ q σj2 ,k τ, Yr pτ q dτ k“1 0. r is a local P-martingale. The latter can be achieved by appealing to the fact the r P-distribution of the process s ÞÑ Yr psq, 0 ď s ď t, coincides with the PM ptq distribution of the process s ÞÑ Xpsq, 0 ď s ď t. Then we choose a Brownian ´ ¯ r r , which r r motion Bpsq, possibly on an extension of the probability space Ω, F, P 287 Download free eBooks at bookboon.com.

(52) Advanced stochastic processes: Part II 288 4. STOCHASTIC DIFFERENTIAL EQUATIONS Stochastic differential equations. ´ ¯ ¯ ş ´ r P r such that N r F, r psq “ s σ τ, Yr pτ q dBpτ r q. For details see we call again Ω, 0 the proof of the implication (ii) ùñ (iii) of Theorem 4.17. With such a Brownian motion we obtain: żs ´ ¯ r r q r σ τ, Y pτ q dBpτ Y psq “ x ` 0 żs ´ żs ´ ¯ ´ ¯ ¯ r r r ` σ τ, Y pτ q c τ, Y pτ q dτ ` b τ, Y pτ q dτ. (4.145) 0. 0. Since. „ ˆ żt ´ ˙ȷ ż ¯ 1 t ˇˇ ´ r ¯ˇˇ2 r r r E exp ´ c s, Y psq dBpsq ´ “1 (4.146) ˇc s, Y psq ˇ ds 2 0 0 ¯ şs ´ r it follows that the process s ÞÑ Bpsq ` 0 c τ, Yr pτ q dτ is a Brownian motion relative to the measure ˙ ȷ „ ˆ żt ´ ż ¯ 1 t ˇˇ ´ r ¯ˇˇ2 r r r r A ÞÑ E exp ´ c s, Y psq dBpsq ´ ˇc s, Y psq ˇ ds , A , A P F. 2 0 0 The equalities in (4.145) and (4.146) complete the proof of Theorem 4.25.. . 3.1. Equations with unique strong solutions possess unique weak solutions. The following theorem shows that stochastic diﬀerential equations with unique pathwise solutions also have unique weak solutions. Its proofs puts the L´evy’s characterization of Brownian motion at work: see Theorem 4.5. 4.31. Theorem. Let the vector and matrix functions bps, xq and σps, xq be as in Theorem 4.25. Fix x P Rd . Suppose that the stochastic (integral) equation żt żt (4.147) Xptq “ x ` σ ps, Xpsqq dBpsq ` b ps, Xpsqq ds 0. 0. possesses unique pathwise solutions. Then this equation has unique weak solutions.. In the proof we employ a certain coupling argument. In fact weak solutions to the equations in (4.3) and (4.4) are recast as two pathwise solutions of the same form as (4.147) on the same probability space. Proof. Let tpBptq : t ě 0q , pΩ, F, Pqu and tpB 1 ptq : t ě 0q , pΩ1 , F1 , P1 qu be two independent Brownian motions. Without loss of generality it is assumed that, for 0 ď t ă 8, ´ ď␣ (¯ Ft “ σ tBpsq : 0 ď s ď tu A P F0 : P rAs “ 0 , and ´ ¯ ď Ft1 “ σ tB 1 psq : 0 ď s ď tu tA1 P F1 : P1 rA1 s “ 0u . `Ť ˘ 1 Moreover, F “ σ tě0 Ft , and a a similar assumption is made for F . Let tXptq : t ě 0u be an adapted process which satisﬁes (4.3), and let tX 1 ptq : t ě 0u. 288 Download free eBooks at bookboon.com.

(53) Advanced stochastic processes: Part II. 3. GIRSANOV TRANSFORMATION. Stochastic differential equations 289. be an adapted process which satisﬁes (4.4). Suppose 0 ď t1 ă t2 ă ¨ ¨ ¨ ă tn ă 8, and let C1 , . . . , Cn be Borel subsets of Rd . We have to prove the equality: P1 rX 1 pt1 q P C1 , . . . , X 1 ptn q P Cn s “ P rX pt1 q P C1 , . . . , X ptn q P Cn s . (4.148). Let pΩ0 , F0 , P0 q be a probability space Ť with a0 Brownian motion tB0 ptq : t ě 0u 0 such that F “ σ ptB0 ptq : t ě 0u tA0 P F : P0 rA0 s “ 0uq. Deﬁne the Rd r0 ptq on Ω ˆ Ω1 ˆ Ω0 as follows: valued processes Y ptq, Y 1 ptq, and B $ 1 ’ pω, ω 1 , ω0 q P Ω ˆ Ω1 ˆ Ω0 ; &Y ptq pω, ω , ω0 q “ X ptq pωq, (4.149) Y 1 ptq pω, ω 1 , ω0 q “ X 1 ptq pω 1 q , pω, ω 1 , ω0 q P Ω ˆ Ω1 ˆ Ω0 ; ’ %B 1 1 r0 ptq pω, ω 1 , ω0 q “ B0 ptq pω0 q , pω, ω , ω0 q P Ω ˆ Ω ˆ Ω0 .. In fact we use the notation Ω0 instead of Ω to distinguish the third component of the space Ω ˆ Ω1 ˆ Ω0 from the ﬁrst. The role of the ﬁrst two components are very similar; the third component is related to the driving Brownian motion tB0 ptq : t ě 0u. The processes Y ptq and ´Y 1 ptq are going to be the pathwise ¯ 1 1 0 r solutions on the same probability space Ω ˆ Ω ˆ Ω0 , F b F b F , Qx : see (4.159) and (4.160) below. On Ω0 the probability measure P0 is determined by prescribing its ﬁnite-dimensional distributions via the equality: P0 rpB0 pt1 q , . . . , B0 ptn qq P Ds “ P rpB pt1 q , . . . , B ptn qq P Ds “ P1 rpB 1 pt1 q , . . . , B 1 ptn qq P Ds .. Brain power. (4.150). By 2020, wind could provide one-tenth of our planet’s electricity needs. Already today, SKF’s innovative knowhow is crucial to running a large proportion of the world’s wind turbines. Up to 25 % of the generating costs relate to maintenance. These can be reduced dramatically thanks to our systems for on-line condition monitoring and automatic lubrication. We help make it more economical to create cleaner, cheaper energy out of thin air. By sharing our experience, expertise, and creativity, industries can boost performance beyond expectations. Therefore we need the best employees who can meet this challenge!. The Power of Knowledge Engineering. Plug into The Power of Knowledge Engineering. Visit us at www.skf.com/knowledge. 289 Download free eBooks at bookboon.com. Click on the ad to read more.

(54) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 290 4. STOCHASTIC. ` d ˘n In (4.150) we have 0 ď t1 ă ¨ ¨ ¨ ă t . n ă 8, and D is a Borel subset of R ` d ˘n 1 Let C be another Borel subset of R . On Ω ˆ Ω0 and Ω ˆ Ω0 the probability measures Qx and Q1x are determined by, respectively, the equalities: Qx rpX pt1 q , . . . , X ptn qq P C, pB0 pt1 q , . . . , B0 ptn qq P Ds. “ P rpX pt1 q , . . . , X ptn qq P C, pB pt1 q , . . . , B ptn qq P Ds ,. and. Q1x rpX 1 pt1 q , . . . , X 1 ptn qq P C, pB0 pt1 q , . . . , B0 ptn qq P Ds. “ P1 rpX 1 pt1 q , . . . , X 1 ptn qq P C, pB 1 pt1 q , . . . , B 1 ptn qq P Ds .. (4.151). Notice that P0 rA0 s “ 0 implies Qx rΩ ˆ A0 s “ Q1x rΩ1 ˆ A0 s “ 0. Consequently, by the Radon-Nikodym’s theorem there are (measurable) functions Qx : F ˆ Ω0 Ñ r0, 1s, and Q1x : F1 ˆ Ω0 Ñ r0, 1s. such that, respectively, ż Qx pA, ω0 q dP0 pω0 q , A P F, A0 P F0 , and Qx rA ˆ A0 s “ żA0 Q1x pA, ω0 q dP0 pω0 q , A1 P F1 , A0 P F0 . (4.152) Q1x rA1 ˆ A0 s “ A0. Here Qx pΩ, ω0 q “ functions. Q1x. pΩ1 , ω0 q “ 1 for P0 -almost all ω0 P Ω0 . Moreover, the. (4.153) ω0 ÞÑ Qx pA, ω0 q , and ω0 ÞÑ Q1x pA, ω0 q 0 are measurable relative to the P0 -completion of F . In addition, the set functions A ÞÑ Qx pA, ω0 q, A P F, and A1 ÞÑ Q1x pA1 , ω0 q, A1 P F1 are P0 -almost surely probability measures. Here we use the fact that, except for negligible sets, the σ-ﬁelds F and F1 are countably determined. Finally, we deﬁne the measure r x : F b F1 b F0 Ñ r0, 1s via the equality Q ż 1 r Qx rA ˆ A ˆ A0 s “ Qx pA, ω0 q Q1x pA1 , ω0 q 1A0 pω0 q dP0 pω0 q “ E0 rω0 ÞÑ Qx pA, ω0 q Q1x pA1 , ω0 q 1A0 pω0 qs .. (4.154). 1 0 Here A, A1 ,!and A0 belong ) to F, F , and F respectively. First we prove that r x. r0 ptq : t ě 0 is Brownian motion with respect to the measure Q the process B r x . From the proof of Theorem The corresponding expectation is written as E 4.5 (i.e., L´evy’s characterization of Brownian motion) it follows that it suﬃces to show that the following equality holds: ” ´ ⟨ ı ⟩¯ ˇ r x exp ´i ξ, B r0 ptq ´ B r0 psq ˇ Fs b F1 b F0 E s s ˆ ˙ 1 “ exp ´ |ξ|2 pt ´ sq , t ą s ě 0, ξ P Rd . (4.155) 2. By deﬁnition Fs “ σ pBpρq : 0 ď ρ ď sq. Similar deﬁnitions are employed for Fs1 and for the σ-ﬁeld Fs0 . In order to prove (4.155) we pick A P Fs , A1 P Fs1 , and A0 P Fs0 . Then by (4.154) we get ” ´ ⟨ ı ⟩¯ r x exp ´i ξ, B r0 ptq ´ B r0 psq 1AˆA1 ˆA0 E 290 Download free eBooks at bookboon.com.

(55) Advanced stochastic processes: II 3. Part GIRSANOV TRANSFORMATION. “. ż. AˆA1 ˆA0. Stochastic differential 291 equations. ´ ⟨ ⟩¯ rx r0 ptq ´ B r0 psq exp ´i ξ, B dQ. “ E0 rω0 ÞÑ exp p´i ⟨ξ, B0 ptq pω0 q ´ B0 psq pω0 q⟩q ˆQx pA, ω0 q Q1x pA1 , ω0 q 1A0 pω0 qs .. (4.156). The process pω0 , tq ÞÑ B0 ptq pω0 q is a Brownian motion relative to P0 , and the events A, A1 , and A0 belong to Fs , Fs1 , and Fs0 respectively, and hence the variable B0 ptq ´ B0 psq is P0 -independent of the variable ω0 ÞÑ Qx pA, ω0 q Q1x pA1 , ω0 q 1A0 pω0 q .. Therefore (4.156) implies ” ´ ⟨ ı ⟩¯ r x exp ´i ξ, B r0 ptq ´ B r0 psq 1AˆA1 ˆA0 E ż ż 1 1 Qx pA, ω0 q Qx pA , ω0 q dP0 pω0 q ˆ exp p´i ⟨ξ, B0 ptq ´ B0 psq⟩q dP0 “ A0 ˆ ˙ 1 2 1 r “ Qx rA ˆ A ˆ A0 s exp ´ |ξ| pt ´ sq . (4.157) 2. The equality in (4.155) is a consequence of (4.157). Since, by definition (see (4.150)). P0 rpB0 pt1 q , . . . , B0 ptn qq P Cs “ P rpB pt1 q , . . . , B ptn qq P Cs ` ˘n for 0 ď t1 ă ¨ ¨ ¨ ă tn ă 8, C Borel subset of Rd , and since the tBptq : t ě 0u is a Brownian motion relative to P, the same is true process tB0 ptq : t ě 0u relative to P0 . Next we compute the quantity: ˇȷ „ˇ żt żt ˇ ˇ r x ˇY ptq ´ x ´ σ ps, Y psqq dB r0 psq ´ b ps, Y psqq dsˇ E ˇ ˇ 0 0 ˇ żt ż ˇ żt ˇ ˇ ˇ “ ˇXptq ´ x ´ σ ps, Xpsqq dBpsq ´ b ps, Xpsqq dsˇˇ dP “ 0. 0. (4.158) process for the. (4.159). 0. Similarly we have ˇȷ „ˇ żt żt ˇ 1 ˇ 1 1 r r ˇ Ex ˇY ptq ´ x ´ σ ps, Y psqq dB0 psq ´ b ps, Y psqq dsˇˇ 0 0 ˇ żt żt ż ˇ ˇ ˇ 1 1 1 “ ˇˇX ptq ´ x ´ σ ps, X psqq dB psq ´ b ps, Xpsqq dsˇˇ dP1 “ 0. (4.160) 0. 0. r x -almost From (4.159) and (4.160) we infer that the following equalities hold Q surely: żt żt r0 psq ` b ps, Y psqq ds and (4.161) Y ptq “ x ` σ ps, Y psqq dB 0 0 żt żt 1 1 r Y ptq “ x ` σ ps, Y psqq dB0 psq ` b ps, Y 1 psqq ds. (4.162) 0. 0. ! ) r x. r Moreover, the process B0 ptq : t ě 0 is a Brownian motion relative to Q From the pathwise uniqueness and the equalities (4.161) and (4.162) we see. 291 Download free eBooks at bookboon.com.

(56) Advanced stochastic processes: Part II DIFFERENTIAL EQUATIONS Stochastic differential equations 292 4. STOCHASTIC. r x -almost surely, that, Q. Y ptq “ Y 1 ptq,. t ě 0.. `. Let 0 ď 0 ă t1 ă ¨ ¨ ¨ ă tn ă 8, and let C be a Borel subset of R (4.163) it follows that r x rpY pt1 q , . . . , Y ptn qq P Cs “ Q r x rpY 1 pt1 q , . . . , Y 1 ptn qq P Cs . Q. ˘ d n. (4.163) . From (4.164). r x shows that the following Using (4.164) and the deﬁnition of the measure Q identities are self-explanatory: r x rpY pt1 q , . . . , Y ptn qq P Cs “ Qx rpX pt1 q , . . . , X ptn qq P C, Ω0 s Q. “ P rpX pt1 q , . . . , X ptn qq P C, Ωs “ P rpX pt1 q , . . . , X ptn qq P Cs .. (4.165). r x is given in (4.154). Similarly we conclude The deﬁnition of the measure Q r x rpY 1 pt1 q , . . . , Y 1 ptn qq P Cs “ P1 rpX 1 pt1 q , . . . , X 1 ptn qq P Cs . Q. (4.166). From (4.165), (4.166), and (4.164) we obtain. P rpX pt1 q , . . . , X ptn qq P Cs “ P rpX 1 pt1 q , . . . , X 1 ptn qq P Cs .. (4.167). The equality in (4.167) implies that the ﬁnite-dimensional distributions of the solution in equation in (4.3) are the same as those of the solution of equation (4.4). So that stochastic diﬀerential equations with unique pathwise solutions also possess unique weak (or distributional) solutions. This concludes the proof of Theorem 4.31.. . 4.32. Example (Tanaka’s example). Let the process t ÞÑ Bptq, t ě 0, be onedimensional Browmian motion on the probability space şt pΩ, F, Pq, and let the continuous process t ÞÑ Xptq be such that Xptq “ 0 sgn pXpsqq dBpsq. Here y sgnpyq “ for y ‰ 0, and sgnpyq “ 0, when y “ 0. It can be proved that such |y| a process exists. If t ÞÑ Xptq solves this equation, then the process t ÞÑ ´Xptq is a solution as well. So we see that the equation dXptq “ sgn pXptqq dBptq, Xp0q “ 0, does not have pathwise unique solutions. On şthe other hand the t process t ÞÑ Xptq is (local) martingale, and, since Bptq “ 0 sgn pXpsqq dXpsq, we get żt t “ ⟨B, B⟩ ptq “ |sgn pXpsqq|2 d ⟨X, X⟩ psq “ ⟨X, X⟩ ptq. 0. Hence, ⟨X, X⟩ ptq “ t. L´evy’s martingale characterization of Brownian motion (see Corollary 4.7 and Theorem 4.5) then implies that the process t ÞÑ Xptq is a Brownian motion on pΩ, F, Pq. So that the distribution of Xptq is that of Brownian motion. Consequently, the equation dXptq “ sgn pXptqq dBptq has unique weak solutions. For more details on Tanaka’s example and its connection with local time see, e.g., Øksendal [106]. Conclusion. In this chapter we treated several aspects of the theory of stochastic diﬀerential equations: strong and weak solutions, L´evy’s characterization. 292 Download free eBooks at bookboon.com.

(57) Advanced stochastic processes: Part II. 3. GIRSANOV TRANSFORMATION. Stochastic differential equations 293. of Brownian motion, exponential martingales, Hermite polynomials with applications to exponential martingales, a version of the martingale representation theorem, and the Girsanov or the Cameron-Martin-Girsanov transformation.. 293 Download free eBooks at bookboon.com. Click on the ad to read more.

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(59) Advanced stochastic processes: Part II. Some related results. CHAPTER 5. Some related results In this section we will discuss, among other things, Fourier transforms of distributions of random variables, positive-definite functions, Bochner’s theorem, L´evy’s continuity theorem, weak convergence of measures, ergodic theorems, projective limits of distributions, Markov processes with one initial probability measure, Doob-Meyer decomposition theorem based on Komlos’ theorem. 1. Fourier transforms Since we will also need signed measures, we will discuss them first. 1.1. Signed measures. Let M “ M pRν , Cq be the vector space of all complex Borel measures on Rν , and let M` be the convex cone of all positive finite Borel measures op Rν . Then we have M “ M` ´M` `i pM` ´ M` q. Thus, every complex Borel measure µ on Rν can be written as µ “ µ1 ´µ2 `i pµ3 ´ µ4 q, where µ1 , µ2 , µ3 and µ4 are finite positive Borel measures. In fact the measures µj , 1 ď j ď 4, can be chosen in the following manner: µ1 pBq “ sup tRe µpCq : C Ď B, C Borel u ;. µ2 pBq “ sup t´Re µpCq : C Ď B, C Borel u ; µ3 pBq “ sup tIm µpCq : C Ď B, C Borel u ;. µ4 pBq “ sup t´Im µpCq : C Ď B, C Borel u .. For this choice of the measures µ1 , µ2 , µ3 and µ4 , the measures µ1 and µ2 and also the measures µ3 and µ4 are mutually singular in the sense that for certain Borel subsets B1 and B3 the following equalities hold: µ1 pBq “ Re µ pB X B1 q , µ3 pBq “ Re µ pB X B3 q ,. µ2 pBq “ Re µ pB X B1c q ; µ4 pBq “ Re µ pB X B3c q .. This decomposition is known under the name Hahn decomposition. In addition, we introduce the variation of a complex measure µ. This measure is denoted as |µ|. It is the bounded positive measure defined by + # ÿ |µpAj q| : A Ě Aj and Aj X Ak “ H for k “ j . |µ| pAq “ sup (5.1) j. Here A is a Borel subset of Rν and the same is true for the elements of the partition Aj , j P N. The norm }µ} of the complex Borel measure µ is then defined by the equality: }µ} “ |µ| pRν q. Supplied with this norm M is turned into a Banach space. By the Riesz representation theorem the space M can 295. 295 Download free eBooks at bookboon.com.

(60) Advanced stochastic processes: Part II 296 5. SOME RELATED RESULTS. Some related results. be taken as the topological dual of the space C0 pRν q, being the Banach space consisting of those complex continuous functions f : Rν Ñ C with the property that limxÑ8 f pxq “ 0. Then C0 pRν q is a closed subspace of the space Cb pRν q, the space of all bounded continuous functions on Rν , which is a Banach space relative to the supremum-norm }¨}8 , given by }f }8 “ supxPRν |f pxq|, f P Cb pRν q. 5.1. Definition. A complex Radon measure on a locally compact space E is a complex Borel measure with the property that for every ϵ ą 0 and every Borel subset B there exists a compact subset K Ă B with the property that |µ pBzKq| ă ϵ. 5.2. Theorem (Riesz). Let E be a locally compact Hausdorﬀ space, which is σcompact, and let Λ : C0 pEq Ñ C be a continuous linear functional. Then there exists a şunique complex Radon measure µ on the Borel ﬁeld of E such that Λpf q “ f dµ, f P C0 pEq. In addition, }Λ} “ }µ} “ |µ| pEq. If Λ is positive in the sense that f ě 0 implies Λpf q ě 0, then the corresponding measure µ is positive as well and }Λ} “ µpEq. Proof. For a proof the reader is referred to the literature. In fact the following construction can be used. Let the measures µj , 1 ď j ď 4 be determined by µ1 pOq “ sup tRe Λpf q : 0 ď f ď 1O , f P C0 pEqu ;. µ2 pOq “ sup t´Re Λpf q : 0 ď f ď 1O , f P C0 pBqu ;. µ3 pOq “ sup tIm Λpf q : 0 ď f ď 1O , f P C0 pEqu ;. µ4 pOq “ sup t´Im Λpf q : 0 ď f ď 1O , f P C0 pEqu ,. where O is any open subset of E. Then it can be shown that, for each 1 ď j ď 4, the set function µj extends to a genuine positive Borel measure on E. This extension is again called µj . Moreover, ˙ ˆż ż ż ż Λpf q “ f dµ1 ´ f dµ2 ` i f dµ3 ´ f dµ4 , f P C0 pEq.. For details the reader is referred to, e.g., [136]. This completes the proof of Theorem 5.2. 5.3. Definition. Let µ be a complex Borel measure on Rν . Then the equality ż µ ppxq “ exp p´i ⟨x, y⟩q dµpyq, x P Rν ,. defines the Fourier transform of the measure µ.. 5.4. Proposition. Let µ be a complex measure on Rν with the property that its Fourier transform is identically zero. Then the measure µ “ 0.. Proof. Let ν be an arbitrary other complex Borel measure on Rν with the property that |p ν pxq| ď 1, x P Rν . Then the following equality holds: ż ż µ ppxqdνpxq “ νppyqdµpyq. 296 Download free eBooks at bookboon.com.

(61) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Hence,. Some related results 297. ˇ "ˇż * ˇ ˇ ν ˇ ˇ }µ} “ sup ˇ f pyqdµpyqˇ : f P Cb pR q , }f }8 ď 1 ˇ * "ˇż ˇ ˇ ν ˇ ˇ “ sup ˇ νppyqdµpyqˇ : |p ν pyq| ď 1, y P R ˇ * "ˇż ˇ ˇ ν “ sup ˇˇ µ ppxqdνpxqˇˇ : |p ν pyq| ď 1, y P R “ 0.. This completes the proof of Proposition 5.4.. . 5.5. Definition. Let φ : Rν Ñ C be a complex valued function. This function is called positive-definite if for every n-tuple of complex numbers λ1 , . . . , λn together with every choice of n vectors ξ p1q , . . . , ξ pnq in Rν , the following inequality holds: n ÿ ` ˘ λj λk φ ξ pjq ´ ξ pkq ě 0, j,k“1. and this for all n P N.. Challenge the way we run. EXPERIENCE THE POWER OF FULL ENGAGEMENT… RUN FASTER. RUN LONGER.. RUN EASIER…. READ MORE & PRE-ORDER TODAY WWW.GAITEYE.COM. 1349906_A6_4+0.indd 1. 22-08-2014 12:56:57. 297 Download free eBooks at bookboon.com. Click on the ad to read more.

(62) Advanced stochastic processes: Part II 298 5. SOME RELATED RESULTS. Some related results. 5.6. Proposition. Let µ be a complex Borel measure on Rν with Fourier transform µ p. Then the following assertions are true: (a) (b) (c) (d). the following inequality holds: |p µpxq| ď }µ}. pp0q “ }µ} are valid. If µ is positive, then the equalities µpRν q “ µ If µ is positive, then the function µ p is positive-definite. The function µ p is uniformly continuous.. Proof. The proof is left as an exercise to the reader.. . 5.7. Definition. Define for µ and ν measures in M, the convolution-product µ ˚ ν via the equalities: ż ż µ ˚ νpBq “ 1B px ` yqdµpxqdνpyq ` ˘ “ µ b ν S ´1 B “ µ b ν tpx, yq P Rν : x ` y P Bu .. Here B is a Borel subset of Rν and S is the (sum) mapping S : px, yq ÞÑ x ` y. Let x P Rν . Define thee Dirac-measure δx by δx pBq “ 1B pxq, B Borel subset ˇ is of Rν . Instead of δ0 it is more customarily to write δ. Let µ P M. Then µ ν ν defined by µ ˇpBq “ µp´Bq, where B is a Borel subset of R . Let f : R Ñ C be a complex function, which is defined on all of Rν . The function fˇ is given by fˇpxq “ f p´xq, x P Rν .. 5.8. Definition. A complex Banach algebra pA, }¨}q is a complex Banach space, endowed with a product which is compatible with the norm. The latter means that the product pa, bq ÞÑ ab, a, b in A, which is a bilinear operation, is continuous in both variables simultaneously. In fact it is assumed that }ab} ď }a} }b} for all a and b in A.. Examples of Banach algebras are the vector spaces C0 pRν q and Cb pRν q, equipped with the supremum-norm and the pointwise multiplication. Let LpXq be the vector space of all continuous linear operators on the Banach space X, supplied with the operator norm and the composition as product. Then LpXq is a noncommutative Banach algebra. The following theorem says that M, supplied with the convolution product, constitutes a (complex) commutative Banach algebra with identity δ. Recall that M stands for the space of all complex Borel measures on Rν . 5.9. Theorem. The normed vector space pM, }¨}q supplied with the convolution product ˚ is a commutative complex Banach algebra with identity δ. If µ and ν belong to M, then the following equalities hold: p q { µ `ν “µ p ` νp, ax µ “ ap µ, µ z ˚ν “µ pνp, µ q“µ p. Here a is a complex number.. Proof. The proof is left as an exercise for the reader.. . The Banach space L1 pRν q can be considered as a closed subspace of MpRν q. This can be done via the following inclusion-mapping: f ÞÑ µf , f P L1 pRν q.. 298 Download free eBooks at bookboon.com.

(63) Advanced stochastic processes: Part II. 1. FOURIER TRANSFORMS. Some related results 299. ş Here µf is the complex measure B ÞÑ B f pxqdx, B P B “ BpRν q, where B is the Borel ﬁeld of Rν . Let µf “ µf,1 ´ µf,2 ` i pµf,3 ´ µf,4 q be the Hahn-Jordan decomposition of the measure µf . Then the following equalities hold: ż ż |µf | pBq “ |f pxq| dx; µf,1 pBq “ max pRe f pxq, 0q dx; B B ż ż µf,2 pBq “ max p´Re f pxq, 0q dx; µf,3 pBq “ max pIm f pxq, 0q dx; B B ż µf,4 pBq “ max p´Im f pxq, 0q dx. B. 5.10. Theorem. Let C00 pRν q be the space of all complex continuous functions with compact support. Then C00 pRν q is a dense subspace of L1 pRν q for the topology of convergence in mean.This means that C00 pRşν q is dense in L1 pRν q relative to the topology generated by the L1 -norm: }f }1 “ |f pxq| dx, f P L1 pRν q.. Proof. Let ϵ ą 0 and let f ě 0 belong to L1 pRν q. It suﬃces that there ş exists a function g P C00 pRν q such that |f pxq ´ gpxq| dx ď ϵ. Since f “ sup 2 nPN. ´n. n. t2 f u “ sup 2. ´n. nPN. n2 ÿn. j“1. 1tf ěj2´n u. we only need to show that, for every pair of positive integers j and n, with 1 ď j ď n2n , there exists a function uj,n P C00 pRν q such that ż ˇ ˇ ˇ1tf ěj2´n u pxq ´ uj,n pxqˇ dx ď ϵ . (5.2) 2n Because assume that the functions uj,n , 1 ď j ď n2n , satisfy (5.2). Then we write fn “ 2´n tminpn, f q2n u and choose n P N so large that ż 1 0 ď pf pxq ´ fn pxqq dx ď ϵ. 2 Then we have ˇ n ż ˇˇ n2 ˇ ÿ ˇ ˇ uj,n pxqˇ ˇf pxq ´ 2´n ˇ ˇ j“1 ż n2 ÿn ż ˇ ˇ ´n ˇ1tf ěj2´n u pxq ´ uj,n pxqˇ dx ď |f pxq ´ fn pxq| dx ` 2 j“1. n2 ÿn. 1 1ϵ “ ϵ. ď ϵ ` 2´n 2 2n j“1. (5.3). Let λ be the ν-dimensional Lebesgue measure. The inequality in (5.2) can be proved by employing the following identities: λpBq “ inf tλpU q : U Ě B, U open u “ sup tλpKq : K Ď B, K compact u (5.4) together with Tietsche’s theorem, which, among other things, says that with a given open subset U and given compact subset K, with K Ă U , there exists a. 299 Download free eBooks at bookboon.com.

(64) Advanced stochastic processes: Part II 300. Some related results. 5. SOME RELATED RESULTS. function u P C00 pRν q with the property that 1K ď u ď 1U . The equalities in (5.4) follow via an argument about Dynkin systems. . This completes the proof of Theorem 5.10. 5.11. Proposition. Let f belong to L1 pRν q. Then ż lim |f px ` yq ´ f pxq| dx “ 0.. (5.5). yÑ0. Proof. By theorem 5.10 it suﬃces to prove (5.5) for f P C00 pRν q. Such a function f is uniformly continuous. Let K be the support of the function f P C00 pRν q. Fix ϵ ą 0 and choose δ ą 0 in such a way that λ pK ` Bpδqq. sup. xPK,yPBpδq. |f px ` yq ´ f pxq| ă ϵ.. Here the symbol Bpδq stands for Bpδq “ δBp1q “ tx P Rν : |x| ď δu. Then we have ż |f px ` yq ´ f pxq| dx ď ϵ. . for |y| ď δ. So the proof of Proposition 5.11 is complete now.. 5.12. Theorem (Riemann-Lebesgue). Let f P L1 pRν q. Then lim fppxq “ 0. xÑ8. ş Of course here we write fppxq “ exp p´i ⟨x, y⟩q f pyqdy.. This e-book is made with. SETASIGN. SetaPDF. PDF components for PHP developers. www.setasign.com 300 Download free eBooks at bookboon.com. Click on the ad to read more.

(65) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 301. Proof. By translation invariance of the Lebesgue-measure we get the equal˙˙ ˆ ˆ ż x 1 p dy. (5.6) exp p´i ⟨x, y⟩q f pyq ´ f y ` π 2 f pxq “ 2 |x| From (5.6) the inequality: ˙ˇ ˆ ż ˇ ˇ ˇ x ˇˇ ˇ p ˇ 1 ˇˇ f pyq ´ f y ` π 2 ˇ dy. (5.7) ˇf pxqˇ ď 2 ˇ |x|. ity:. A combination of (5.7) and Proposition 5.11 yields the desired result, and completes the proof of Theorem 5.12. 5.13. Theorem (Stone-Weierstrass). Let E be a locally compact Hausdorﬀ space and let A be a subalgebra of C0 pEq, which separates points of E and which is closed under complex conjugation. That is, if f belongs to A, then f also belongs to A. Then A is dense in C0 pEq. Proof. Let E △ be the one-point compactiﬁcation (Alexandroﬀ compactiﬁcation) and A1 “ A ‘ C1 “ tf ` λ1 : f P A, λ P Cu. Here 1 is the constant function with value 1 and functions f P A vanish in △. The theorem of StoneWeierstrass, applied to the compact Hausdorﬀ space E △ results in the desired result, and completes the proof of Theorem 5.13. ! ) 5.14. Theorem. The set fp : f P C00 pRν q is a subalgebra of C0 pRν q that is closed under taking complex conjugates. This algebra is dense in C0 pRν q with the supremum-norm. ) ! Proof. The fact that the set A :“ fp : f P L1 pRν q is a subalgebra of C0 pRν q follows from the standard properties of the Fourier transform in combip nation with Theorem 5.12. Since fp “ fˇ it also follows that this algebra is closed under complex conjugation. In order to apply the Theorem of Stone-Weierstrass we still have to show that A separates the points of Rν . To this end take x0 and y0 ‰ x0 P Rν . Then there exists a bounded open neighborhood V in Rν such that exp p´i ⟨x0 , y⟩q ´ exp p´i ⟨y0 , y⟩q “ 0 for y P V . Next consider the function f : y ÞÑ pexp pi ⟨x0 , y⟩q ´ exp pi ⟨y0 , y⟩qq vpyq, where v is a function in C00 pRν q with v ě 1V . Then we see ż fppx0 q ´ fppy0 q “ |exp p´i ⟨x0 , y⟩q ´ exp p´i ⟨y0 , y⟩q|2 vpyqdy ą 0. (5.8). From (5.8) it immediately follows that A separates the points of Rν . The assertion in Theorem 5.14 now follows from Theorem 5.13. . In the following theorem we collect some properties of positive-deﬁnite functions. 5.15. Theorem. Let φ : Rν Ñ C be a positive-deﬁnite function. Then φ possesses the following properties: (a) φp´xq “ φpxq, x P Rν ; (b) |φpxq| ď φp0q, P Rν ;. 301 Download free eBooks at bookboon.com.

(66) Advanced stochastic processes: Part II 302 5. SOME RELATED RESULTS. Some related results. ν (c) |φpxq ´ φpyq|2 ď 2φp0q pφp0q ´ Re φpx ´ yqq, x, y P˘R ; ` ` ˘ 2 2 2 2 (d) φp0q |φpx ` yqφp0q ´ φpxqφpyq| ď φp0q ´ |φpxq| φp0q2 ´ |φpyq|2 .. Proof. Fix x and y in Rν and consider the matrices ¨ ˛ ˆ ˙ φp0q φpxq φpyq φp0q φp´xq φp0q φpx ´ yq‚. en ˝φpxq φpxq φp0q φp0q φpyq φpx ´ yq. (a) and (b) Since the first one of these two matrices is positive-hermitian it follows that: φp´xq “ φpxq en |φpxq| ď φp0q. (c) Since the second matrix is positive-hermitian, we obtain by the choice of the constants a1 , a2 and a3 : λ |φpxq ´ φpyq| a1 “ 1, a2 “ , a3 “ ´a2 φpxq ´ φpyq the following inequality for all λ P R: ` ˘ φp0q 1 ` 2λ2 ` 2λ |φpxq ´ φpyq| ´ 2λ2 Re φpx ´ yq ě 0. (5.9). The inequality in (c) is a consequence of (5.9).. (d) The determinant of a positive hermitian matrix is non-negative. So that, if the 3 ˆ 3 matrix ˛ ¨ 1 λ µ ˝λ 1 ξ ‚ (5.10) µ ξ 1 is positive-hermitian, then we get the inequality 1 ` λµξ ` λµξ ě |λ|2 ` |µ|2 ` |ξ|2 ,. which is equivalent with ˇ ˇ ` ˘` ˘ ˇξ ´ λµˇ2 ď 1 ´ |λ|2 1 ´ |µ|2 .. (5.11). The inequality in (d) then follows from (5.11) by associating the second matrix with the matrix in (5.10) and by employing (5.11). The proof of Theorem 5.15 is complete now.. . 5.16. Proposition. Let g be a function in L1 pRν q. Then the following equalities hold: spectral radius of pgq “ lim }g ˚n }1{n “ }p g }8 . 1 nÑ8. In the theory of Banach algebras the Beurling-Gelfand formula gives a relationship between the spectral radius and the norm of an element. More precisely, let pA, }¨}q be a Banach algebra with unit e. A Banach algebra is a Banach space with a multiplication px, yq ÞÑ xy which satisfies the usual axioms of distributivity and scalar multiplication. The norm satisfies }xy} ď }x} ¨ }y}, x, y P A, }e} “ 1. By definition, the spectrum σpxq of an element x P A is given by σpxq “ tλ P C : λe ´ x R GpAqu. Here GpAq is the group of invertible. 302 Download free eBooks at bookboon.com.

(67) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 303. elements of A: x P GpAq if and only if there exists a (unique) element y P A such that xy “ yx “ e. Then σpxq is a non-empty compact subset of C contained in the disc of radius }x}: σpxq Ă tλ P C : |λ| ď }x}u. In fact we have the Beurling-Gelfand formula for the spectral radius: sup |λ| “ lim sup }xn }1{n “ inf }xn }1{n , x P A.. (5.12). nPN. nÑ8. λPσpxq. Let A “ L1 pRν q‘Cδ, where δ is the Dirac measure at zero, with a multiplication given by the convolution product: pf ` αδq ˚ pg ` βδq “ f ˚ g ` αg ` αβ, f, g P L1 pRν q , α, β P C,. 1 ν and with the ş norm given by }f ` αδ} “ }f }L1 ` |α|, f P L pR q , α P C. Here f ˚ gpxq “ f pyqgpx ´ yq dy. Then A is a commutative Banach algebra with unit δ. The spectral radius ρpf q of f P L1 pRν q is given by the supremum norm of its Fourier transform: ˇ ˇ ˇ ˇ 1{n 1{n ρpf q “ lim sup }f ˚n }L1 “ inf }f ˚n }L1 “ sup ˇfppxqˇ ,. ş. nÑ8. nPN. xPRν. where fppxq “ e´ix¨y f pyq dy. The interested reader can find more information in Bonsall and Duncan [22], in Yosida [154], and in several other places like Lax [81]. Proof of Proposition 5.16. For a proof we refer the reader to a book on functional analysis with Banach algebras as a topic. Good references are Rudin [117], Theorem 11.9 together with Example (e), and Folland [55], Theorem 1.30 combined with Theorem 4.2. www.sylvania.com. We do not reinvent the wheel we reinvent light. Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges. An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day.. Light is OSRAM. 303 Download free eBooks at bookboon.com. Click on the ad to read more.

(68) Advanced stochastic processes: Part II 304 5. SOME RELATED RESULTS. Some related results. The following theorem is a very important representation theorem. It will be used in Theorem 5.25 and in the continuity theorem of L´evy: Theorem 5.42. 5.17. Theorem (Bochner). Let φ : Rν Ñ C be a function. The following assertions are equivalent: (i) The function φ is continuous and positive-definite; p. (ii) There exists a positive Borel measure µ op Rν such that φ “ µ. The Borel measure µ in piiq is unique.. Proof. (i) ñ (ii).ş Define the linear functional Λ : M Ñ C by means of the equality: Λpνq “ φpxqdνpxq, ν P M. Define the involution ν ÞÑ νr via the equality: νrpAq “ νp´Aq. Because, by hypothesis, the function φ is positivedefinite we see that the functional Λ is positive in the sense that Λpν ˚ νrq ě 0 for all ν P M: see inequality (5.26) in Proposition 5.23 further on. By CauchySchwartz inequality we then obtain ´ ´ ¯¯1{2 |Λpνq| “ |Λpν ˚ δq| ď pΛ pν ˚ νrqq1{2 Λ δ ˚ δr ď pΛ pν ˚ νrqq1{2 φp0q1{2. (by inductionwith respect to n). ¯¯1{2n`1 ´ ´ řn`1 ´j n φp0q j“1 2 ď Λ pν ˚ νrq˚2 › ›1{2n`1 řn`1 ´j 1{2n`1 › ˚2n › ď }φ}8 φp0q j“1 2 . ›pν ˚ νrq ›. (5.13). By letting n tend to 8 in (5.13) we deduce › ›1{2n`1 › ˚2n › |Λpνq| ď lim inf ›pν ˚ νrq › φp0q nÑ8 a “ spectral radius of ν ˚ νrφp0q.. By applying (5.13) and (5.14) to a measure ν of the form νpBq “ where f belongs to L1 pRν q we obtain ˇż ˇ b ˇ ˇ ˇ φpxqf pxqdxˇ ď spectral radius of f ˚ frφp0q. ˇ ˇ. (5.14) ş. B. f pxqdx, (5.15). ş In (5.15) we wrote frpxq “ f p´xq and f ˚ gpxq “ f pyqgpx ´ yqdy, for f and g belonging to L1 pRν q. Next we realize that L1 pRν q, equipped with the L1 -norm and the convolution product ˚, is a Banach-algebra and that the spectral radius of an L1 -function f is given by the supremum-norm the Fourier transform of f : see Proposition 5.16. From (5.15) we infer ˇż ˇ d›ˇ ˇ› › › ˇ ›ˇ{ˇ› ˇ ˇ φpxqf pxqdxˇ ď ›ˇf ˚ frˇ› φp0q ď ››fp›› φp0q. (5.16) ˇ ›ˇ ˇ ˇ› 8. 8. 304 Download free eBooks at bookboon.com.

(69) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 305. ! ) ş Next deﬁne Λ0 : fp : f P L1 pRν q Ñ C via the equality Λ0 pfpq “ φpxqf pxqdx,. f P L1 pRν q. From (5.16) it follows that the functional Λ0 has a unique extension as a continuous linear! functional, which ) we call again Λ0 , on the uniform 1 ν p closure of the subalgebra f : f P L pR q . By the Stone Weierstrass theorem (Theorem 5.14) this closure coincides with C0 pRν q. The Riesz representation theorem applies there exists a bounded Borel measure µ such ş to the eﬀect that 1 p p that Λ0 pf q “ f pxqdµpxq, f P L pRν q. From this it follows that ż. φpxqf pxqdx “ Λ0 pfpq “. ż. fppyqdµpyq “. ż. ppxqf pxq dx. µ. p. The function φ being positive-deﬁnite it follows that the Consequently, φ “ µ measure µ is positive. This proves the implication (i) ùñ (ii).. (ii) ñ (i). Let µ be a ﬁnite positive Borel measure. Then its Fourier transp is a uniformly continuous positive-deﬁnite function. The proof of these form µ assertions is left to the reader. . The proof of Theorem 5.17 is complete now.. An alternative proof runs as follows: the idea is taken from Theorem 5.10 in L˝orinczi et al [88]. We need the following lemmas. 5.18. Lemma. Let φ : Rν Ñ C be a (uniformly) continuous positive-definite 2 1 function, and fix t ą 0. Then the function ξ ÞÑ e´ 2 t|ξ| φpξq is also (uniformly) continuous and positive-definite. Proof. Let ξj , 1 ď j ď n, belong to Rν , and let λj , 1 ď j ď n, be complex numbers. Then n ÿ. j,k“1. 2. 1. λj λk e´ 2 t|ξj ´ξk | φ pξj ´ ξk q. 1 “ `? ˘ν 2πt. ż. Rν. n ÿ. j,k“1. 2. λj eiξj ¨y λk eiξk ¨y φ pξj ´ ξk q e´|y|. {p2tq. The claim in Lemma 5.18 follows from (5.17).. dy ě 0.. (5.17) . 5.19. Lemma. Let ψ : Rν Ñ C be a function which belongs to L1 pRν q, and let V1 be a boundedş open neighborhood of the origin in Rν . Put Vn “ nV1 , n P N. Let m pVn q “ 1Vn pξq dξ “ nν m pV1 q be the Lebesgue measure of Vn . Then, uniformly in x P Rν , ş ş ż eipξ´ηq¨x ψ pξ ´ ηq dξ dη . (5.18) eiξ¨x ψpξq dξ “ lim Vn Vn nÑ8 m pVn q Rν. 305 Download free eBooks at bookboon.com.

(70) Advanced stochastic processes: Part II 306 5. SOME RELATED RESULTS. Some related results. Proof. By employing standard properties, like translation invariance and the homothety property of the Lebesgue measure, we deduce the following equalities: ş ş ż eipξ´ηq¨x ψ pξ ´ ηq dξ dη V n Vn iξ¨x e ψpξq dξ ´ m pVn q Rν ş ş ż eiξ¨x ψ pξq dξ dη Vn Vn ´η iξ¨x e ψpξq dξ ´ “ m pVn q Rν ş ş ş ş iξ¨x e ψ pξq dξ dη eiξ¨x ψ pξq dξ dη Vn Rν zpVn ´ηq V1 Rν zpnV1 ´nηq “ . (5.19) “ m pVn q m pV1 q From (5.19) we infer ˇż ˇ ş ş ş ş ipξ´ηq¨x ˇ ˇ |ψ pξq| dξ dη e ψ pξ ´ ηq dξ dη V1 Rν zpnV1 ´nηq ˇ ˇ . eiξ¨x ψpξq dξ ´ Vn Vn ˇ ˇď ˇ Rν ˇ m pVn q m pV1 q (5.20) Hence, by using the Lebesgue’s dominated convergence theorem the equality in (5.18) is readily established. Moreover, this limit is uniform in x P Rν . This completes the proof of Lemma 5.19. . 360° thinking. .. function which 5.20. Lemma. Let ψ : Rν Ñ C be a continuous positive-definite ş belongs to L1 pRν q. Then, for all x P Rν the inequality Rν eiξ¨x ψpξq dξ ě 0 holds.. 360° thinking. .. 360° thinking. .. Discover the truth at www.deloitte.ca/careers. © Deloitte & Touche LLP and affiliated entities.. Discover the truth at www.deloitte.ca/careers. Deloitte & Touche LLP and affiliated entities.. © Deloitte & Touche LLP and affiliated entities.. Discover the truth 306 at www.deloitte.ca/careers Click on the ad to read more Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities.. Dis.

(71) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 307. Proof. Since the function ψ is positive-definite and continuous the righthand side of (5.18) is non-negative. So the assertion in Lemma 5.20 follows from Lemma 5.19. 5.21. Lemma. Let ψ : Rν Ñ C be a continuous positive-definite function which belongs to L1 pRν q, and let µ beş a bounded complex-valued Borel measure on Rν with Fourier transform µ ppxq “ Rν e´ix¨y dµpyq. The the following equality holds: ż ż ż 1 ψpξq dµpξq “ eiξ¨x ψpξq dξ µ ppxq dx. (5.21) ν p2πq Rν Rν Rν If φ : Rν Ñ C is an arbitrary continuous positive-definite function, and if µ is a bounded complex-valued Borel measure on Rν , then ż ż ż 2 1 1 φpξq dµpξq “ lim eiξ¨x e´ 2 t|ξ| φpξq dξ µ ppxq dx, (5.22) ν tÓ0 p2πq Rν Rν Rν and ˇż ˇ ˇ ˇ ˇ ˇ µpxq| . (5.23) sup |p ˇ ν φpξq dµpξqˇ ď φp0q xPR ν R. Proof. From Fubini’s theorem we get ż ż 1 eiξ¨x ψpξq dξ µ ppxq dx p2πqν Rν Rν ż ż ż 1 iξ¨x e ψpξq dξ e´ix¨y dµpyq dx “ p2πqν Rν Rν ν R ˙ ż ż ˆ ż 1 iξ¨x “ e ψpξq dξ e´ix¨y dx dµpyq ν p2πq ν ν ν R ż żR R ´1 FF pψq pyq dµpyq “ ψpyq dµpyq, “ Rν. (5.24). Rν. where F denotes the Fourier transform with inverse F´1 . The equalities in (5.24) imply the equality in (5.21). In order to prove he equality in (5.22) we first 2 1 observe that by Lemma 5.18 the functions of the form ξ ÞÑ φt pξq :“ e´ 2 t|ξ| φpξq, t ą 0, are positive-definite and continuous, because φ is so. Applying the equality in (5.21) to the function φt shows ż ż 2 1 e´ 2 t|ξ| φpξq dµpξq φpξq dµpξq “ lim tÓ0 Rν Rν ż ż 2 1 1 “ lim eiξ¨x e´ 2 t|ξ| φpξq dξ µ ppxq dx. (5.25) ν tÓ0 p2πq Rν Rν. The equality in (5.22) follows from (5.25). Finally, the inequality in (5.23) follows from (5.22) and Lemma 5.20. So the proof of Lemma 5.21 is complete now. . Second proof of Theorem 5.17. Let M “ M pRν q be the collection of bounded complex Borel measures on R|nu , and consider the functional ż φpξq dµpξq, µ P M. p ÞÑ Λφ : µ Rν. 307 Download free eBooks at bookboon.com.

(72) Advanced stochastic processes: Part II 308. 5. SOME RELATED RESULTS. Some related results. Then Λφ can be extended to the uniform closure of the collection tp µ : µ P Mu such that |Λφ pf q| ď φp0q }f }8 for all f in this closure. This closure contains all constant functions and all continuous functions on Rν which tend to 0 at 8. By the Riesz representation theorem there exists a positive measure µφ on the Borel field of Rν such that ż ż ż ż φpξq dµpξq “ µ ppxq dµφ pxq “ e´iξ¨x dµpξq dµφ pxq ν ν ν ν R R R R ż ż ż “ e´iξ¨x dµφ pxq dµpξq “ µ pφ pξq dµpξq, Rν. Rν. Rν. for all µ P M. It follows that φpξq “ µ pφ pξq. This completes the proof of the theorem of Bochner: Theorem 5.17. 5.22. Lemma. Let φ : Rν Ñ C be a continuous function, and let µ be a complex Borel measure on Rν with compact support. So |µ| pRν zKq “ 0 for some compact subset K of Rν . Then ˇ+ #ˇż ż n ˇ ˇ ÿ ˇ ˇ aj ak φ pxj ´ xk qˇ “ 0, inf ˇ φpx ´ yqdµpxqdµpyq ´ ˇ ˇ j,k“1 where the infimum is taken over all aj P C, xj P K0 , 1 ď j ď n, n P N, and where K0 is the smallest compact set K with the property that |µ| pRν zKq “ 0.. Proof. Fix ϵ ą 0, and choose a partition pUj : 1 ď j ď nq of K0 with the property that ϵ |φ px ´ yq ´ φ px1 ´ y 1 q| ď , |µ| pK0 q2 x, x1 P Uj and y, y 1 P Uk , and write aj “ µpUj q. Then for xj P Uj , 1 ď j ď n, we have ˇ ˇżż n ˇ ˇ ÿ ˇ ˇ aj ak φ pxj ´ xk qˇ φpx ´ yqdµpxqdµpyq ´ ˇ ˇ ˇ j,k“1 ˇ ˇ ż n ż ˇ ˇÿ ˇ ˇ pφpx ´ yq ´ φ pxj ´ xk qq dµpxqdµpyqˇ “ˇ ˇ ˇj,k“1 Uj Uk ż n ż ÿ |φpx ´ yq ´ φ pxj ´ xk q| d |µ| pxqd |µ| pyq ď j,k“1 Uj. Uk. ż n ż ÿ ϵ d |µ| pxqd |µ| pyq “ ϵ. ď |µ| pK0 q2 j,k“1 Uj Uk. This proves Lemma 5.22.. . 5.23. Proposition. (a) Let φ : Rν Ñ C be a continuous function. The following assertions are equivalent. (i) The function φ is positive-definite; ş (ii) For every function f P C00 pRν q the inequality φpxqf ˚ frpxqdx ě 0 holds; 308 Download free eBooks at bookboon.com.

(73) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 309. (iii) Every Borel measure µ with compact support satisfies the inequality: ż φpxqd pµ ˚ µ rq pxq ě 0. (5.26). (b) If φ is positive-definite and if µ is a bounded complex Borel measure on Rν , then inequality (5.26) in (iii) also holds. ş Proof. (a) (iii) ñ (ii). Choose µ of the form µpBq “ B f pxqdx, with f P C00 pRν q fixed. ř (ii) ñ (i). Let µ be of the form µ “ nj“1 aj δxj . Approximate the de Dirac ş ş measures δxj by measures of the form B ÞÑ B fj,N pxqdx in the sense that ż ż n ÿ lim φpxqd pµN ˚ µĂ φpxqd pµ ˚ µ rq pxq “ aj ak φ pxj ´ xk q . N q pxq “ N Ñ8. Here the measure µN is defined by µN pBq “. řn. j“1. aj. ş. j,k“1. B. fj,N pxqdx, B P BpRν q.. We will turn your CV into an opportunity of a lifetime. Do you like cars? Would you like to be a part of a successful brand? We will appreciate and reward both your enthusiasm and talent. Send us your CV. You will be surprised where it can take you.. 309 Download free eBooks at bookboon.com. Send us your CV on www.employerforlife.com. Click on the ad to read more.

(74) Advanced stochastic processes: Part II 310. 5. SOME RELATED RESULTS. Some related results. (i) ñ (iii). Let µ be a Borel measure of compact support. Then there exists a sequence of measures pµN : N P Nq,where every µN is of the form µN “ řN j“1 aj,N δxj,N and where ż ż φpxqd pµ ˚ µ rq pxq “ lim φpxqd pµN ˚ µĂ N q pxq N Ñ8. “ lim. N Ñ8. N ÿ. j,k“1. aj,N ak,N φ pxj,n ´ xk,N q ě 0.. That such a sequence of measures exists pµN : N P Nq follows from Lemma 5.22. : m P Nq be an increasing sequence of compact subsets of Rν such (b) Let pKm Ť8 ν that R “ m“1 Km and such that Km Ă interior pKm`1 q for all m P N. Since, in addition, ż ż ´ ¯¯ ´ Č pxq φpxqd pµ ˚ µ rq pxq “ lim φpxqd p1Km µq ˚ p1 Km µq mÑ8. assertion (b) follows from the results in (a).. This completes the proof of Proposition 5.23.. . 5.24. Definition. The weak topology (or Bernoulli topology) on M is the locally convex topology σ pM, Cb pRν qq. Let µ0 P M. So that every σ pM, Cb pRν qqneighborhood of µ0 contains a neighborhood of the form ˇż ˇ * n " č ˇ ˇ µ P M : ˇˇ fj d pµ ´ µ0 qˇˇ ă ϵj . (5.27) j“1. Here, the functions f1 , . . . , fn are bounded and continuous, and the numbers ϵ1 , . . . , ϵn are strictly positive. A net pµαş : α P Aq şM converges to the measure µ for the topology σ pM, Cb pRν qq if limα f dµα “ f dµ for all f P Cb pRν q.. We write µ “ weak- limα µα . The space M can also be supplied with the vague topology. This is the locally convex topology σ pM, C00 pRν qq. For the vague topology the functions f1 , . . . , fn in (5.27) are required to şbelong toş C00 pRν q and the net pµα : α P Aq converges to µ P M provided limα f dµα “ f dµ for all f P C00 pRν q. We write µ “ vague- limα µα . Let M` :“ tµ P M : µ ě 0u and let. CP :“ CP pRν q “ tφ P Cb pRν q : φ positive-definiteu .. The following theorem expresses the fact that the set M` , endowed with the weak topology and CP , endowed with the compact-open topology T, are homeomorphic. The compact-open topology is also called the topology of uniform convergence on compact subsets of Rν . So that a net pφα : α P Aq converges to φ, if limα supxPK |φα pxq ´ φpxq| “ 0 for every compact subset K of Rν . 5.25. Theorem. The Fourier transform µ ÞÑ µ p, µ P M` , is a homeomorphism from ` ` ` ` ˘˘ M , σ M , Cb pRν q onto pCP, Tq . 310 Download free eBooks at bookboon.com.

(75) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 311. Proof. Let pµα : α P Aq be a net in M` that weakly converges to µ P M` relative to the weak topology. We will prove that the net pp µα : α P Aq converges uniformly on compact subsets to µ p. Fix ϵ ą 0. Then choose δ ą 0 in such a way that δ p3 ` µpRν qq ă ϵ and choose a function f P C00 pRν q such that ż 0 ď f ď 1 and p1 ´ f qdµ ă δ. Since weak- lim µα “ µ there exists α0 P A such that ż ż ν µα pR q “ 1dµα ă 1dµ ` 1 “ µpRν q ` 1 en for all α ě α0 . Define the zero-neighborhood V by. ż. p1 ´ f q dµα ă δ. V “ tx P Rν : |1 ´ exp p´i ⟨x, y⟩q| ď δ : for all y P supppf qu .. Then for those α P A and those x1 and x2 P Rν which satisfy α ě α0 and x1 ´ x2 P V the following inequalities hold: ż pα px2 q| ď |exp p´i ⟨x1 , y⟩q ´ exp p´i ⟨x2 , y⟩q| dµα pyq |p µα px1 q ´ µ ż ď |1 ´ exp p´i ⟨x1 ´ x2 , y⟩q| f pyqdµα pyq ż ` ˘ ` |1 ´ exp p´i ⟨x1 ´ x2 , y⟩q| 1 ´ f pyq dµα pyq ż ż ` ˘ ď δ f pyqdµα pyq ` 2 1 ´ f pyq dµα pyq ď δ pµpRν q ` 1q ` 2δ ď ϵ.. (5.28). ppx2 q| ď ϵ for x1 and x2 P Rν for which By (5.28) it follows that |p µpx1 q ´ µ x1 ´ x2 P V . Next choose a compact subset K in Rν . Then there exist y1 , . . . , yn Ť n in Rν such that K Ď j“1 pyj ` V q and thee exist αj P A, 1 ď j ď n, such that ppyj q| ď ϵ |p µα pyj q ´ µ. for α ě αj , j “ 1, . . . , n.. Then choose α1 P A in such a way that α1 ě αj for j “ 1, . . . , n. For x P yj ` V and α ě α1 we get ppxq| ď |p µα pxq ´ µ pα pyj q| ` |p µα pyj q ´ µ ppyj q| ` |p µpyj q ´ µ ppxq| ď ϵ |p µα pxq ´ µ. and hence. µα pxq ´ µ ppxq| ď 3ϵ. sup |p xPK. This proves that the Fourier transform is continuous for the indicated topologies. Conversely, suppose that the net pp µα : α P Aq converges uniformly on compact subsets to µ p. Then we will show the following two equalities: (a) lim şµα pRν q “ µpRν q;ş (b) lim φpxqdµα pxq “ φpxqdµpxq for all functions φ P C00 pRν q.. From Theorem 5.26 below it then follows that weak- lim µα “ µ. The equality in (a) follows from: pα p0q “ µ pp0q “ µ pRν q . lim µα pRν q “ lim µ 311 Download free eBooks at bookboon.com.

(76) Advanced stochastic processes: Part II 312 5. SOME RELATED RESULTS. Some related results. Let ϵ ą 0 be arbitrary and let φ P C00 pRν q. Choose a function f P C00 pRν q with the property that › › ϵ › › . ›φ ´ fp› ď 2µ pRν q ` 1 8 Then we infer ˇż ˇ ż ˇ ˇ ˇ φpxqdµα pxq ´ φpxqdµpxqˇ ˇ ˇ ˇ ˇ ˇż ´ ˇż ˇż ´ ˇ ¯ ¯ ˇ ˇ ˇ ˇ ˇ ˇ p p p ˇ ˇ ˇ ˇ ˇ ďˇ f pxq ´ φpxq dµpxqˇˇ φpxq ´ f pxq dµα pxqˇ ` ˇ f pxqd pµα ´ µq pxqˇ ` ˇ ż › › › › ν ν p µα pxq ´ µ ppxq| |f pxq| dx ď ›φ ´ f › pµα pR q ` µ pR qq ` |p 8 ż ϵ pµα pRν q ` µ pRν qq ď ` sup |µα pxq ´ µ ppxq| |f pxq| dx. (5.29) 2µpRν q ` 1 xPsupppf q The inequality. ˇż ˇ ˇ ˇ ˇ lim sup ˇ φpxqd pµα ´ µq pxqˇˇ ď ϵ. α. follows from (5.29). As a consequence we see that (b) is proved now. Together with Theorem 5.26 which follows next this completes the proof of Theorem 5.25. . I joined MITAS because I wanted real responsibili� I joined MITAS because I wanted real responsibili�. Real work International Internationa al opportunities �ree wo work or placements. �e Graduate Programme for Engineers and Geoscientists. Maersk.com/Mitas www.discovermitas.com. �e G for Engine. Ma. Month 16 I was a construction Mo supervisor ina const I was the North Sea super advising and the No he helping foremen advis ssolve problems Real work he helping fo International Internationa al opportunities �ree wo work or placements ssolve pr. 312 Download free eBooks at bookboon.com. Click on the ad to read more.

(77) Advanced stochastic processes: Part II. 1. FOURIER TRANSFORMS. Some related results 313. 5.26. Theorem. A net pµα : α P Aq in M` converges weakly to µ P M` if and only if the net pµα : α P Aq converges vaguely to µ and if lim µα pRν q “ µ pRν q . α. (5.30). Proof. The weak topology is stronger than the vague topology and from weak convergence the equality in (5.30) also follows. Hence, the indicated conditions are necessary. Conversely, let a net pµα : α P Aq converge vaguely M` to µ and assume that (5.30) is satisfied. We will prove that µ is the weak limit of the net pµα : α P Aq. Therefore pick f P Cb pRν q and ϵ ą 0 arbitrary but fixed. Choose a compact subset K such that µ pRν zKq ă ϵ. In addition, choose a function h P C00 pRν q in such a way that 1K ď h ď 1. By these hypotheses the following (in-)equalities hold: ż ż lim p1 ´ hqdµα “ p1 ´ hqdµ ď µ pRν zKq ă ϵ and also lim. ż. f hdµα “. ż. f hdµ.. Hence, there exists an α0 P A such that ż p1 ´ hqdµα ă ϵ en. (for α ě α0 ) ˇż ˇ ˇ ˇ ˇ f hd pµα ´ µqˇ ă ϵ. ˇ ˇ. But then for α ě α0 we get ˇż ˇ ˇ ˇ ˇ f d pµα ´ µqˇ ˇ ˇ ˇ ˇż ˇ ˇż ˇ ˇż ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ď ˇ f hd pµα ´ µqˇ ` ˇ f p1 ´ hqdµˇ ` ˇ f p1 ´ hqdµα ˇˇ. ď ϵ p1 ` 2 }f }8 q , ş ş which shows that lim f dµα “ f dµ.. This completes the proof of Theorem 5.26.. . 5.27. Corollary. The following assertions are true: (a) The set CP is a convex cone, which is closed for the topology of uniform convergence on compact subsets. (b) With φ the functions φ and Re φ also belong to CP . (c) If φ1 and φ2 belong to CP , then the same is true for the product φ1 φ2 . (d) For every y P Rν the function x ÞÑ exp p´i ⟨x, y⟩q belongs to CP . Convex combinations of such functions belong to CP . Proof. The proof is left as an exercise for the reader.. 313 Download free eBooks at bookboon.com. .

(78) Advanced stochastic processes: Part II 314. 5. SOME RELATED RESULTS. Some related results. 5.28. Definition. A function ψ : Rν Ñ C is called negative-definite if for all n P N and for all complex numbers a1 , . . . , an and for all vectors xp1q , . . . , xpnq in Rν the inequality n ´ ` ˘ ÿ ` pjq ˘¯ pjq pkq pkq aj ak ψ x ` ψ px q ´ ψ x ´ x ě0 (5.31) j,k“1. holds. The symbol CN denotes the collection of all continuous negative-definite functions on Rν . If ψ belongs to CN , then the same is true for ψ andRe ψ. The collection CN is a convex cone. If ψ belongs to CN , then ψp0q ě 0 and ψpxq “ ψp´xq for all x P Rν . A function ψ is negative-definite if and only if ψ has the following properties: (1) ψp0q ě 0; (2) For every x P Rν the equality ψpxq “ ψp´xq holds; (3) For every numbers a1 , . . . , an , for řnn P N and for every n-tuple of complex p1q which j“1 aj “ 0, and for all vectors x , . . . , xpnq in Rν the following inequality holds: n ÿ. j,k“1. ` ˘ aj ak ψ xpjq ´ xpkq ď 0.. If the function ψ is negative-definite, then so is the function ψ ´ ψp0q. If φ is positive-definite, then the function φp0q ´ φ is negative-definite. The following theorem establishes an important connection between negativeand positive-definite functions. 5.29. Theorem (Schoenberg). A function ψ belongs to CN if and only the following two conditions are satisfied: (i) ψp0q ě 0; (ii) For every t ą 0 the function exp p´tψq is continuous and positivedefinite. Let ψ be a negative-definite function. Then, by Bochner’s theorem together with the theorem of Schoenberg, there exists for every t ą 0 a sub-probability pt “ exp p´tψq. We return to this measure µt on the Borel field of Rν such that µ aspect when we discuss the notion convolution semigroup of measures.. Proof. First that CN . Let xp1q , . . . , xpnq belong to Rν . ` pjqsuppose ` pkq ˘ ˘ ψ belongs ` pjq to ˘ Write aj,k “ ψ x `ψ x ´ψ x ´ xpkq . Then the matrix with entries aj,k is positive hermitian. But then the matrix with entries exp`paj,k`q is also ˘˘ positive hermitian. Let a1 , . . . , an belong to C and write a1j “ exp ´ψ xpjq aj . Then we see n n ÿ ÿ ` ` pjq ˘˘ pkq exp ´ψ x ´ x exppaj,k qa1j a1k ě 0. (5.32) aj ak “ j,k“1. j,k“1. 314 Download free eBooks at bookboon.com.

(79) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 315. From (5.32) it follows that the function exp p´ψq is then positive-definite. The same procedure can be repeated for the function tψ. Conversely, if (i) and (ii) are satisfied, then, for every t ą 0, the function ψt :“ 1 ´ exp p´tψq “ 1 ´ exp p´tψp0qq ` exp p´tψp0qq ´ exp p´tψq is negative-definite. But then the ψt function ψ is negative-definite as well, because ψ “ lim . Since tÓ0 t 1 ´ exp p´tψpxqq , ψpxq “ şt ds exp p´sψpxqq 0 for t ą 0 but small enough, we see that the function ψ is continuous at x. So the proof of Theorem 5.29 is now complete.. . 5.30. Definition. A family of Borel measures pµt : t ě 0q with the following properties: (a) µt pRν q ď 1 for t ą 0; (b) µs ˚ µtş “ µs`t for ş all s and t ě 0; (c) limtÓ0 f dµt “ f dµ0 “ f p0q “ δ0 pf q for all f P C00 pRν q;. is called a (vaguely continuous) convolution semigroup of measures on Rν .. 315 Download free eBooks at bookboon.com. Click on the ad to read more.

(80) Advanced stochastic processes: Part II 316 5. SOME RELATED RESULTS. Some related results. The following theorem says that a vaguely continuous convolution semigroups is in fact everywhere weakly continuous. 5.31. Theorem. There exists a one-to-one correspondence between vaguely continuous semigroups of measures and negative-definite functions. (a) If pµt : t ě 0q is a vaguely continuous convolution semigroup of measures, then there exists a unique continuous negative-definite function pt “ exp p´tψq, for all t ě 0. ψ such that µ (b) Conversely, if ψ is a negative-definite function, then there exists a vaguely continuous convolution semigroup of measures pµt : t ě 0q such that µ pt “ exp p´tψq for all t ě 0. Of course, this semigroup is unique.. Proof. (a) Define, for t ą 0, the function ψ via the equality pt 1´µ . ψ“ żt µ ps ds. (5.33). 0. Since µ pşs µ pt “ µ ps`t we see that ψ does not depend on the choice of t. Put t gptq “ 0 µ ps ds. Then we see that gp0q “ 0 and gptqψ ` g 1 ptq “ 1, and hence 1 ´ exp p´tψq . From the latter it follows that µ pt “ exp p´tψq. The gptq “ ψ Theorem of Schoenberg (Theorem 5.29) implies then that the function ψ is negative-definite. The functions µ ps , s ě 0, are continuous. So the same is true for ψ. (b) Since ψ is a negative-definite function, the functions exp p´tψq are positivedefinite by the theorem of Schoenberg. The theorem of Bochner (Theorem pt “ 5.17) yields the existence of sub-probability measures pµt : t ě 0q such that µ exp p´tψq. Since p0 pξq pt pξq “ lim exp p´tψpξqq “ 1 “ µ lim µ tÓ0. tÓ0. ş Theorem 5.43 in the next section implies that limtÓ0 f dµt “ f p0q for functions f P C00 pRν q. The proof of Theorem 5.31 is now complete.. . 5.32. Remark. In the proof of Theorem 5.31 part (a) there is a problem if żt şt 1 ps ds vanishes somewhere. However, notice that lim µ ps ds “ the integral 0 µ tÓ0 t 0 µ p0 pointwise. şt It follows that, certainly, for t “ tpξq ą 0 small enough, the ps pξqds “ 0. This fact can be used to circumvent this problem. expression 0 µ. 5.33. Remark. In the proof of Theorem 5.31 part (b) Theorem 5.43 of the next section was employed. This can şbe averted as well. ş Therefore consider fp, with f P L1 pRν q. Then limtÓ0 fppxqdµt pxq “ limtÓ0 f pxqp µt pxqdx “ ş ş p p f pxqdx “ f şp0q “ f dµ0 . By the ştheorem of Stone-Weierstrass from this we obtain limtÓ0 f pxqdµt pxq “ f p0q “ f pxqdµ0 pxq. 316 Download free eBooks at bookboon.com.

(81) Advanced stochastic processes: Part II. 1. FOURIER TRANSFORMS. Some related results 317. 5.34. Proposition. Let pµt : t ě 0q be a vaguely continuous semigroup of Borel measures on Rν . Suppose that all these measures are probability measures. Then the following assertions hold: (a) weak- limtÑt0 ,tą0ˇşµt “ µt0 for all t0 ě ˇ ş 0; ˇ (b) limtÑt0 supxPRν f px ´ yqdµt pyq ´ f px ´ yqdµt0 pyqˇ “ 0 for all t0 P r0, 8q and for all functions f P C0 pRν q. Proof. (a) First we look at µt pRν q ´ µt0 pRν q “ exp p´tψp0qq ´ exp p´t0 ψp0qq . It follows that lim µt pRν q ´ µt0 pRν q “ 0.. tÑt0. For the same reason we see that pt pξq “ lim exp p´tψpξqq “ exp p´t0 ψpξqq “ µ pt0 pξq. lim µ. tÑt0. tÑt0. By using theorem 5.43 in the next section we see that weak- lim µt “ µt0 . tÑt0 ,tą0. Of course, in this proof the function ψ denotes the negative-definite function from Theorem 5.31. (b) Let g P C0 pRν q be of the form g “ fp with f P L1 pRν q. Then we see ˇ ˇż ż ˇ ˇ ˇ fppx ´ yqdµt pyq ´ fppx ´ yqdµt0 pyqˇ ˇ ˇ ˇ ˇżż ˇ ˇ pt0 p´zqq exp p´i ⟨x, z⟩q f pzqdz ˇˇ “ ˇˇ pp µt p´zq ´ µ ż ď |exp p´tψp´zqq ´ exp p´t0 ψp´zqq| |f pzq| dz ż ď |exp p´ |t ´ t0 | ψp´zqq ´ 1| |f pzq| dz. (5.34). The assertion in (b) now follows from (5.34) together with the theorem of StoneWeierstrass, and completes the proof of Proposition 5.34. 5.35. Proposition. Let pµt : t ě 0q be a vaguely continuous semigroup of probability measures on the Borel field of Rν . Define for every n-tuple t1 , . . . , tn with 0 ď t1 ă ¨ ¨ ¨ ă tn , the probability measure Pt1 ,...,tn on the Borel field of pRν qn via de formula Pt1 ,...,tn pBq. “ µt1 b µt2 ´t1 b ¨ ¨ ¨ b µtn ´tn´1 ppx1 , . . . , xn q P pRν qn : Vn px1 , . . . , xn q P Bq ż ż (5.35) “ dµt1 px1 q . . . dµtn ´tn´1 pxn q1B pVn px1 , . . . , xn qq ,. 317 Download free eBooks at bookboon.com.

(82) Advanced stochastic processes: Part II 318 5. SOME RELATED RESULTS. Some related results. where B is a Borel subset of pRν qn and where Vn : pRν qn Ñ pRν qn is the linear mapping given by: Vn : px1 , x2 , . . . , xn q ÞÑ px1 , x1 ` x2 , . . . , x1 ` ¨ ¨ ¨ ` xn q. Then the family tppRν qn , B pRν qn , Pt1 ,...,tn q : pt1 , . . . , tn q P r0, 8qn , n P Nu forms a projective system of probability measures. ` ˘ Proof. Let B P B pRν qn and let B 1 P B pRν qn`1 be defined by ␣ ( B 1 “ pz1 , . . . , zn`1 q P pRν qn`1 : pz1 , . . . , zk , zk`2 , . . . , zn`1 q P B .. Let t1 ă ¨ ¨ ¨ ă tk ă s ă tk`1 ă ¨ ¨ ¨ ă tn be an pn ` 1q-tuple of increasing times. We have to prove the following equality: Pt1 ,...,tk ,s,tk`1 ,...,tn pB 1 q “ Pt1 ,...,tn pBq. Since the vector Vn`1 py1 , y2 , . . . , yn`1 q :“ py1 , y1 ` y2 , . . . , y1 ` ¨ ¨ ¨ ` yn`1 q. belongs to B 1 if and only if the vector. py1 , y1 ` y2 , . . . , y1 ` ¨ ¨ ¨ ` yk , y1 ` ¨ ¨ ¨ ` yk`2 , . . . , y1 ` ¨ ¨ ¨ ` yn`1 q belongs to B, we get what follows: Pt1 ,...,tk ,s,tk`1 ,...,tn pB 1 q “ µt1 b ¨ ¨ ¨ b µtk ´tk´1 b µs´tk b µtk`1 ´s b ¨ ¨ ¨. b µtn ´tn´1 tpy1 , . . . , yn`1 q : Vn`1 py1 , . . . , yn`1 q P B 1 u ż ż ż ż ż “ dµt1 py1 q . . . dµtk ´tk´1 pyk q dµs´tk pyq dµtk`1 ´s pzq dµtk`2 ´tk`1 pzk`2 q . . . ż dµtn ´tn´1 pzn q1B 1 pVn`1 py1 , . . . , yk , y, z, zk`2 , . . . , zn qq ż ż ż ż ż “ dµt1 py1 q . . . dµtk ´tk´1 pyk q dµs´tk pyq dµtk`1 ´s pzq dµtk`2 ´tk`1 pzk`2 q . . . ż dµtn ´tn´1 pzn q1B pVn py1 , . . . , yk , y ` z, zk`2 , . . . , zn qq. relative to µs´tk bµtk`1 ´s and use the equality ş ş(apply Fubini’s theorem, integrate gpy ` zqdµu pyqdµv pzq “ g pzk`1 q dµu`v pzk`1 q) ż ż ż ż “ dµt1 py1 q . . . dµtk ´tk´1 pyk q dµtk`1 ´tk pzk`1 q dµtk`2 ´tk`1 pzk`2 q ż . . . dµtn ´tn´1 pzn q 1B pVn py1 , . . . , yk , zk`1 , . . . , zn qq ż ż “ dµt1 py1 q . . . dµtn ´tn´1 pyn q1B py1 , . . . , y1 ` . . . ` yn q “ Pt1 ,...,tn pBq.. 318 Download free eBooks at bookboon.com.

(83) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 319. This proves the required equality in case 1 ď k ď n ´ 2. The other cases, which are tn´2 ă s ă tn´1 , tn´1 ă s ă tn , tn ă s and t1 ą s, are left as an exercise for the reader. . So the proof of Proposition 5.35 is complete now.. 5.36. Proposition. Let pµt : t ě 0q be a vaguely continuous semigroup of probability measures on the Borel field of Rν . Define, for every n-tuple t1 , . . . , tn the probability measure Pt1 ,...,tn , where t1 ă ¨ ¨ ¨ ă tn , as in Proposition 5.35. Then there exists a unique probability measure P on the product field of pRν qr0,8q such that P ppXpt1 q, . . . , Xptn qq P Bq “ Pt1 ,...,tn pBq , for all Borel subsets B of pRν qn . Likewise there exists, for every x P Rν , a unique probability measure Px on the product field of pRν qr0,8q such that Px ppXpt1 q, . . . , Xptn qq P Bq “ P ppx ` Xpt1 q, . . . , x ` Xptn qq P Bq ż “ dµt1 px1 q b ¨ ¨ ¨ b dµtn ´tn´1 pxn q1B px ` x1 , . . . , x ` x1 ` . . . ` xn q ,. for all Borel subsets B of pRν qn .. Here the state variable Xptq : pRν qr0,8q Ñ Rν is defined by Xptqpωq “ ωptq, where ω belongs to the product pRν qr0,8q .. no.1. Sw. ed. en. nine years in a row. STUDY AT A TOP RANKED INTERNATIONAL BUSINESS SCHOOL Reach your full potential at the Stockholm School of Economics, in one of the most innovative cities in the world. The School is ranked by the Financial Times as the number one business school in the Nordic and Baltic countries.. Stockholm. Visit us at www.hhs.se. 319 Download free eBooks at bookboon.com. Click on the ad to read more.

(84) Advanced stochastic processes: Part II 320. 5. SOME RELATED RESULTS. Some related results. Proof of Proposition 5.36. Apply Kolmogorov’s extension theorem. 5.37. Theorem. Let pµt : t ě 0q be a vaguely continuous semigroup of probability measures on the Borel field of Rν . Define for every n-tuple t1 , . . . , tn the probability measure Pt1 ,...,tn , where t1 ă ¨ ¨ ¨ ă tn , as in Proposition 5.35 and let Px , x P Rν , be the unique probability measure on the product field of Ω “ pRν qr0,8q such that Px ppXpt1 q, . . . , Xptn qq P Bq ż ż “ dµt1 px1 q . . . dµtn ´tn´1 pxn q1B px ` x1 , . . . , x ` x1 ` . . . ` xn q .. (5.36). Let Fs be the σ-field on Ω generated by Xpuq, 0 ď u ď s. For t ą s the variable Xptq ´ Xpsq is independent of Fs and Xptq ´ Xpsq possesses the same Px -distribution as Xpt ´ sq ´ x, which is µt´s . Proof. Fix t ą s, let f : pRν qn Ñ R be a Borel measurable function, and suppose that 0 ď s1 ă ¨ ¨ ¨ ă sn “ s. Let g : Rν Ñ R be another bounded Borel measurable function. Then the following equalities hold true: E pf pXps1 q, . . . , Xpsn qq g pXptq ´ Xpsqqq ż ż ż “ dµs1 px1 q . . . dµsn ´sn´1 pxn q dµt´s pxqf px1 , . . . , x1 ` . . . ` xn q gpxq “ E pf pXps1 q, . . . , Xpsn qqq E pg pXptq ´ Xpsqqq .. Now let H be the vector space of Fs -measurable bounded random variables Y with the property that E pY gpXptq ´ Xpsqqq “ E pY q E pgpXptq ´ Xpsqqq. Then H satisﬁes the hypotheses of Lemma 5.100. Whence, H contains all bounded Fs measurable random variables. Since, in addition, the function g is an arbitrary bounded continuous function, it follows that the state variable Xptq ´ Xpsq is independent of Fs . This completes the proof of Theorem 5.37. 5.38. Theorem. Let pΩ, F, Pq be a probability space and let pXptq : t ě 0q be a family of state variables with state space Rν . Assume that these state variables are measurable relative to the σ-fields F and BpRν q. Suppose that lim E rf pXptqqs “ f p0q for all f P C00 pRν q, tÓ0. and also that for every t ą s the variable Xptq ´ Xpsq is independent of the σfield σ pXpuq : 0 ď u ď sq and that Xptq ´ Xpsq possesses the same distribution as Xpt ´ sq. Then the mapping B ÞÑ µt pBq :“ P pXptq P Bq defines a vaguely continuous semigroup of probability measures on Rν . Proof. It is clear that every ş measure µt is a probability measureν is on ν the Borel σ-ﬁeld of R . Since f dµt “ E pf pXptqqq, for f P C00 pR q, the equality limtÓ0 Epf pXptqqq “ f p0q, entails that the family pµt : t ě tq is vaguely continuous at 0. The convolution property still has to be proved. It suﬃces to µt pξq “ µ ps`t pξq for all s and t ě 0, and for all ξ P Rν . To this prove that µ ps pξqp end consider ż ż exp p´i ⟨ξ, x⟩q dµs pxq. exp p´i ⟨ξ, y⟩q dµt pyq. 320 Download free eBooks at bookboon.com.

(85) Advanced stochastic processes: Part II. 1. FOURIER TRANSFORMS. Some related results 321. “ E pexp p´i ⟨ξ, Xpsq⟩qq E pexp p´i ⟨ξ, Xptq⟩qq (the variable Xptq has the same distribution as Xps ` tq ´ Xpsq) “ E pexp p´i ⟨ξ, Xpsq⟩qq E pexp p´i ⟨ξ, Xps ` tq ´ Xpsq⟩qq (Xps ` tq ´ Xpsq does not depend on Xpsq) “ E pexp p´i ⟨ξ, Xpsq ` Xps ` tq ´ Xpsq⟩qq “ E pexp p´i ⟨ξ, Xps ` tq⟩qq “ µ ps`t pξq.. Since 0 “ Xp0q ´ Xp0q it follows that µ0 has the distribution δ0 . This proves Theorem 5.38. 5.39. Definition. Let pΩ, F, Pq be a probability space and let the mapping X : pt, ωq ÞÑ Xpt, ωq “ Xptqpωq satisfy the hypotheses mentioned in Theorem 5.38. (So that for t ą s the state variable Xptq ´ Xpsq does not depend on the σ-ﬁeld σ pXpuq : 0 ď u ď sq and Xptq ´ Xpsq possesses the same distribution as Xpt ´ sq; moreover, the equality limsÓ0 E pf pXpsqqq “ f p0q holds for all f P C0 pRν q). Then the process X is called a L´evy-process, that begins at Xp0q “ 0. Important L´evy-processes are the Poisson process with jumps 1 and the Brownian motion. The one-dimensional distributions of a Poisson process X (with jumps 1 and of intensity λ) are given by pλtqk exp p´λtq, k P N. k! For details on Poisson processes see Subsection 5.4 in Chapter 1. The Brownian motion B (with drift 0, intensity I and which starts in 0) possesses as onedimensional distributions: ¸ ˜ ż |y|2 1 dy. exp ´ P pBptq P Bq “ a 2t p2πtqν B P pXptq “ kq “. For more details on Brownian motion see the Section 4 in Chapter 1 and Section 3 in Chapter 2. In addition, see Chapter 3. A L´evy-process with initial distribution µ is a family of F-B-measurable mappings Xptq : Ω Ñ Rν such that Xp0q has the distribution µ, and such that the process t ÞÑ Xptq ´ Xp0q is a L´evy-process that starts at 0. If the initial distribution µ “ δx , then it said that the process X starts at x. If X “ pXptq : t ě 0q is a L´evy-process that starts at 0, then px ` Xptq : t ě 0q is a L´evy-process, which starts at x. The Poisson process Xj (with jumps 1 and intensity λ) which starts at j P N possesses as marginal or one-dimensional distributions: pλtqk´j exp p´λtq1r0,8q pk ´ jq, k P N. k! Thus the distributions of the processes pXj ptq : t ě 0q and pj ` Xptq : t ě 0q, where X is the Poisson-process which starts at 0, are the same. The Brownian P pXj ptq “ kq “. 321 Download free eBooks at bookboon.com.

(86) Advanced stochastic processes: Part II 322 5. SOME RELATED RESULTS. Some related results. motion Bx (with drift 0, intensity I and which starts at x) possesses the following one-dimensional distributions: ¸ ˜ ż 1 |x ´ y|2 dy. exp ´ P pBx ptq P Bq “ a 2t p2πtqν B 5.40. Definition. Let E be a locally compact Hausdorﬀ space and let tP ptq : t ě 0u. be a family of linear operators of C0 pEq to the space L8 pE, Eq. Here E is the Borel ﬁeld of E. This family is called a Feller semigroup, or Feller-Dynkin semigroup provided it possesses the following properties: (i) (ii) (iii) (iv) (v). semigroup-property: P ps ` tq “ P psqP ptq and P p0q “ I; positivity preserving: f ě 0, f P C0 pEq, implies P ptqf ě 0; contractive: 0 ď f ď 1, f P C0 pEq, implies 0 ď P ptqf ď 1; continuity: limtÓ0 rP ptqf s pxq “ f pxq for all f P C0 pEq and for all x P E; invariance: P ptqC0 pEq Ď C0 pEq for all t ě 0.. In the presence of (i), (v) and (iii) assertion (iv) is equivalent with (iv1 ) limtÑt0 ,tą0 }P ptqf ´ P pt0 qf }8 “ 0 for all f P C0 pEq.. 322 Download free eBooks at bookboon.com. Click on the ad to read more.

(87) Advanced stochastic processes: Part II 1. FOURIER TRANSFORMS. Some related results 323. 5.41. Theorem. Let pΩ, F, Pq be a probability space, and pXptq : t ě 0q be a family of state variables with state space Rν . Suppose that these state variables are measurable relative to the σ-fields F and BpRν q. In addition, suppose that limtÓ0 E pf pXptqqq “ f p0q for all f P C00 pRν q and also that for every t ą s the variable Xptq ´ Xpsq does not depend on the σ-field σ pXpuq : 0 ď u ď sq, an d this for all t ą s ě 0. Moreover, by hypothesis, the variable Xptq ´ Xpsq has the same distribution as Xpt ´ sq. Define the operator P ptq from L8 pRν q to of P ptq to itself by rP ptqf s pxq “ E pf px ` Xptqqq, f P L8 pRν q. The restriction ! ) ˇ ν ν ˇ C0 pR q leaves the space C0 pR q invariant, and the family P ptq C0 pRν q : t ě 0 is a Feller semigroup (also called a Feller-Dynkin semigroup). Proof. It is clear that every operator P ptq is contractive and positivity preserving. It is also clear that limtÓ0 rP ptqf s pxq “ f pxq for all x P Rν and for all f P C0 pRν q. We still have to prove the invariance property. Let f “ gp, where g belongs to L1 pRν q. Then we obtain rP ptqf s pxq “ E pf px ` Xptqqq “ E pp g px ` Xptqqq ż “ exp p´i ⟨ξ, x⟩q E pexp p´i ⟨ξ, Xptq⟩qq gpξqdξ.. (5.37). By the lemma of Riemann-Lebesgue (Theorem 5.12), the equalities in (5.37) imply the equality lim rP ptqf s pxq “ 0. xÑ8. The continuity of the function P ptqf is clear as well. As a consequence, P ptq maps the space tp g : g P L1 pRν qu to C0 pRν q. The theorem of Stone-Weierstrass implies that the space tp g : g P L1 pRν qu is dense in C0 pRν q for the uniform topology. Because of the contractive character of the operator P ptq it then follows that P ptq leaves the space C0 pRν q invariant. In order to ﬁnish we prove the semigroup-property. Again we take the Fourier transform gp of a function g P L1 pRν q and we consider rP ps ` tqp g s pxq. “ E pp g px ` Xps ` tqqq ż “ e´i⟨ξ,x⟩ E pexp p´i ⟨ξ, Xps ` tq⟩qq gpξqdξ ż “ e´i⟨ξ,x⟩ E pexp p´i ⟨ξ, Xps ` tq ´ Xpsq⟩q exp p´i ⟨ξ, Xpsq⟩qq gpξqdξ. (the variable Xps ` tq ´ Xpsq is independent of Xpsq) ż “ e´i⟨ξ,x⟩ E pexp p´i ⟨ξ, Xps ` tq ´ Xpsq⟩qq E pexp p´i ⟨ξ, Xpsq⟩qq gpξqdξ (the variable Xps ` tq ´ Xpsq has the same distribution as Xptq) ż “ e´i⟨ξ,x⟩ E pexp p´i ⟨ξ, Xptq⟩qq E pexp p´i ⟨ξ, Xpsq⟩qq gpξqdξ “ E pω ÞÑ E pω 1 ÞÑ gp px ` Xpsqpωq ` Xptq pω 1 qqqq . 323 Download free eBooks at bookboon.com. (5.38).

(88) Advanced stochastic processes: Part II 324. 5. SOME RELATED RESULTS. Some related results. The semigroup-property then follows from (5.38) together with the Theorem of Stone-Weierstrass which, among other things, implies that the space ( ␣ gp : g P L1 pRν q is dense in C0 pRν q for the uniform topology.. This completes the proof of Theorem 5.41.. . 2. Convergence of positive measures We begin with the continuity theorem of L´evy. 5.42. Theorem (L´evy). Let pµn : n P Nq be a sequence of bounded positive Borel measures on Rν . Assume that there exists a function φ : Rν Ñ C, which is continuous at 0, such that lim µ pn pxq “ φpxq. nÑ8. for all x P Rν . Then there exists a bounded positive Borel measure such that weak- lim µn “ µ. nÑ8. Proof. The function φ is a point-wise limit of positive-definite functions and so it is itself positive-definite as well. Since the function φ is continuous at 0, inequality (c) in Theorem 5.15 implies the continuity of φ. By Bochner’s theorem (Theorem 5.17) there exists a positive bounded Borel measure µ such that pn pxq “ φpxq “ µ ppxq (5.39) lim µ nÑ8. for all x P R . Next, let f be an arbitrary function in C00 pRν q. Then we see ˇż ˇż ˇ ˇ ż ˇ ˇ ˇ ˇ ppxqqˇˇ dx lim sup ˇˇ fpdµn ´ fpdµˇˇ “ lim sup ˇˇ f pxq pp µn pxq ´ µ nÑ8 nÑ8 ż ppxq| dx. (5.40) µn pxq ´ µ ď lim sup |f pxq| |p ν. nÑ8. In view of (5.39) we see that the integrand in (5.40) converges pointwise to 0. µn p0q ` µ pp0qq. Then c is finite and |f pxq| |p µn pxq ´ µ ppxq| is Write c “ supnPN pp 1 dominated by the L -function c |f pxq|. By the dominated convergence theorem (Lebesgue) it follows from (5.40) that ˇż ˇ ż ˇ ˇ lim sup ˇˇ fpdµn ´ fpdµˇˇ “ 0. (5.41) nÑ8. ) ! Since the subspace fp : f P C00 pRν q is uniformly dense in C0 pRν q (see Theş ş orem 5.14), from (5.41) it follows that limnÑ8 φdµn “ φdµ for all functions φ P C00 pRν q. Theorem 5.26 then implies weak- limnÑ8 µn “ µ. This proves the continuity theorem of L´evy: Theorem 5.42. . 324 Download free eBooks at bookboon.com.

(89) Advanced stochastic processes: Part II. 2. CONVERGENCE OF POSITIVE MEASURES. Some related results 325. In the following theorem we compare several equivalent forms of weak convergence. If a “ pa1 , . . . , aν q and b “ pb1 , . . . , bν q belong to Rν and if aj ă bj , 1 ď j ď ν, then we write a ă b and also pa, bs “ pa1 , b1 s ˆ ¨ ¨ ¨ ˆ paν , bν s.. 5.43. Theorem. Let pµα : α P Aq be a directed system (a net) in M` consisting of sub-probability measures (so that µα pRν q ď 1, α P A) and let µ P M` be a sub-probability measure as well. Let pfk : k P Nq be a sequence in C0 pRν q with a linear span which is dense in C0 pRν q. The following assertions are then equivalent:. (1) The net pµα : α P Aq converges weakly to µ; (2) For every bounded Borel measurable function f ş: Rν Ñ şC which is continuous in µ-almost all points the equality limα f dµα “ f dµ holds; (3) The net pµα : α P Aq converges vaguely to µ and lim µα pRν q “ µ pRν q; (4) For every closed subset F of Rν the inequality lim supα µα pF q ď µpF q holds and lim µα pRν q “ µ pRν q ; α. (5) For every open subset G of Rν the inequality lim inf α µα pGq ě µpGq holds and lim µα pRν q “ µ pRν q ; α ˆ ˙ ˝ ν (6) For every Borel subset B of R , for which µ BzB “ 0, the equality. limα µα pBq “ µpBq holds; (7) For every pair of points pa, bq P Rν ˆ Rν such that aj ă bj , 1 ď j ď ν, where a “ pa1 , . . . , aν q, b “ pb1 , . . . , bν q, with the property that µ tx P Rν : `xj “‰ aj u `“ µ ‰tx P Rν : xj “ bj u “ 0, j “ 1, . . . , ν, the ν ν equality limα µα a, b “ µ a, b holds ş and limşα µα pR q “ µ pR q. ν (8) For every k P N the equalities limα fk dµα “ fk dµ and limα µα pR q “ µ pRν q hold; pα pxq “ µ ppxq holds. (9) For every x P Rν the equality limα µ (10) For every a P Rν for which µ tx P Rν : xj “ aj u “ 0, j “ 1, . . . , ν, the equality lim µα rp´8, a1 s ˆ ¨ ¨ ¨ ˆ p´8, aν ss “ µ rp´8, a1 s ˆ ¨ ¨ ¨ ˆ p´8, aν ss α. holds and limα µα pRν q “ µ pRν q. Proof. The equivalence of the assertions (1) and (9) is a consequence of Theorem 5.25. The equivalence of (1) and (3) is a consequence of Theorem 5.26. The implication (1) ñ (8) is trivial. Theş implication ş (8) ñ (3) can be proved as follows. From (8) it follows that limα φdµα “ φdµ for all φ in the linear span of pfk : k P Nq Y t1u. So that for f P C0 pRν q ` C1 and φ in the span of pfk : k P Nq Y t1u we see that ˇ ˇż ˇ ˇ ˇ lim sup ˇ f d pµα ´ µqˇˇ α. 325 Download free eBooks at bookboon.com.

(90) Advanced stochastic processes: Part II 326. Some related results. 5. SOME RELATED RESULTS. ˇż ˇ ˇż ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ď lim sup ˇ pf ´ φq d pµα ´ µqˇ ` lim sup ˇ φd pµα ´ µqˇˇ α α. ď }f ´ φ}8 lim sup pµα pRν q ` µ pRν qq α. “ 2 }f ´ φ}8 µ pRν q .. (5.42). Assertion (3) follows because the linear span of pfk : k P Nq Y t1u is uniformly dense in C0 pRν q`C1. From the previous arguments it follows that the assertions (1), (3), (8) and (9) are equivalent. (2) ñ (1). This implication is trivial. (1) ñ (4). Let F be a closed subset of Rν . Choose a sequence of functions puj : j P Nq in Cb pRν q in such a way that 1F ď uj`1 ď uj ď 1, j P N, and such that 1F pxq “ limjÑ8 uj pxq for all x P Rν . Then the equality ż ż lim sup µα pF q ď inf lim sup uj dµα “ inf uj dµ “ µpF q α. jPN. jPN. α. holds. This proves assertion (4) starting from (1). (4) ô (5). These implications are easy to verify.. 326 Download free eBooks at bookboon.com. Click on the ad to read more.

(91) Advanced stochastic processes: Part II 2. CONVERGENCE OF POSITIVE MEASURES. ˆ. ˝. (5) ñ (6). Let B be a Borel subset of R such that µ BzB ν. (5), what is equivalent to (4), it follows that. Some related results 327. ˙. “ 0. Then, from. ˆ ˙ ˝ ` ˘ ` ˘ lim sup µα pBq ď lim sup µα B ď µ B “ µ B α ˆ ˙ ˝ ď lim inf µα B ď lim inf µα pBq . α. α. (5.43). Hence, limα µα pBq “ µpBq. (6) ñ (1). Let 0 ď f ď 1 be a continuous function. Because ż1 ż ż1 µ tf ě ξu dξ “ µ tf ą ξu dξ f dµ “ 0. 0. ş1. we see that 0 µ tf “ ξu dξ “ 0. Thus for almost all ξ the equality µ tf “ ξu “ 0 follows. For a certain sequence pαℓ : ℓ P Nq in A, we then obtain by (6) the following (in-)equalities ż1 ż ż1 tf ą ξu dµ “ µ tf ě ξu dξ f dµ “ 0 0 ż1 ż1 lim µα tf ě ξu dξ “ lim µα tf ą ξu dξ “ 0. 1 ď n 2. α. 2n ÿ. k“1. 0. ␣. lim µα f ą k2 α. ´n. (. α. 2n ␣ ( 1 ÿ 1 1 lim µαℓ f ą k2´n ` n ` n ď n 2 2 k“1 ℓÑ8 2. (Fatou’s lemma) ż. 2n 1 1 ÿ ď lim 1tf ěk2´n u dµαℓ ` n n ℓÑ8 2 k“1 2 ż ż 1 1 ď lim f dµαℓ ` n “ lim inf f dµα ` n . α ℓÑ8 2 2. From (5.44) it then follows that, always for 0 ď f ď 1, ż ż f dµ ď lim inf f dµα , α. and also. ż. ż. p1 ´ f qdµ ď lim inf p1 ´ f qdµα . α. (5.44). (5.45). (5.46). Since, in addition, limα µα pRν q “ µ pRν q we see by (5.45) and (5.46) that ş limα f dµα “ f dµ for every function f P Cb pRν q for which 0 ď f ď 1. Since the linear span of such functions coincides with Cb pRν q assertion (1) follows from (6). (5) ñ (2). Let f be a real-valued bounded function which µ-almost everywhere continuous. Without loss of generality we assume that 0 ď f ď 1. 327 Download free eBooks at bookboon.com.

(92) Advanced stochastic processes: Part II 328 5. SOME RELATED RESULTS. Some related results. (otherwise replace f with af ` b, with a and b appropriately chosen constants). Then define the functions f X and f Y respectively by f X pxq “ inf sup f pyq en f Y pxq “ sup inf f pyq. U PUpxq yPU. U PUpxq yPU. (5.47). ş It follows that pf X ´ f Y q dµ “ 0 and also f Y ď f ď f X . Hence, for an appropriately chosen sequence pαℓ : ℓ P Nq, ż ż 2n ( 1 1 ÿ ␣ Y Y µ f ą k2´n f dµ “ f dµ ď n ` n 2 2 k“1 2n ␣ ( 1 ÿ 1 lim inf µαℓ f Y ą k2´n ď n` n 2 2 k“1 ℓÑ8 n. 2 ␣ ( 1 ÿ 1 µαℓ f Y ą k2´n ď n ` lim inf n ℓÑ8 2 2 ż k“1 ż 1 1 Y ď n ` lim inf f dµα ď n ` lim inf f dµα . α α 2 2. From (5.48) it follows that. ż. f dµ ď lim inf α. ż. (5.48). f dµα .. ş ş For the same reason the inequality p1 ´ f qdµ ď lim infşα p1 ´ f qdµαş holds. Because, in addition, limα µα pRν q “ µ pRν q we see that f dµ “ limα f dµα . This proves (2) starting from (5). (6) ñ (7). This assertion is trivial. (7) ñ (8). In this part of the proof we write pa, bs for the interval pa1 , b1 s ˆ ¨ ¨ ¨ ˆ paν , bν s, if a and b are` points in` Rν ‰for which aj ă bj for 1 ď j ď ν. From ‰ (7) it follows that limα µα a, b “ µ a, b for all points a and b in Rν with the property that µty P Rν : xj “ aj u “ µty P Rν : xj “ bj u “ 0 for all 1 ď j ď ν. ν Next pick for f a function řnin C00 pR q with values in R. Let g be an arbitrary function of the form g “ j“1 f pxj q1paj ,bj s , where aj and bj are points in Rν with the following properties: µty P Rν : yk “ aj,k u “ µty P Rν : yk “ bj,k u “ 0, for 1 ď k ď ν, and aj,k ă bj,k for j “ `1, . . . ,‰n, and 1 ď k ď ν. In addition, suppose that xj belongs to the “interval” aj , bj . The we get ż ż ż lim sup f dµα ď lim sup pf ´ gqdµα ` lim sup gdµα α α α ż ż ν ν ď }f ´ g}8 µ pR q ` gdµ ď 2 }f ´ g}8 µ pR q ` f dµ. Since f is uniformly continuous we are able to choose, for a given ϵ ą 0, a function g of the form as ş such a way that }f ´ g}8 ď ϵ. This proves ş above in the inequality lim supα f dµα ď f dµ. ş argument can be applied to ş The same the function ´f . It follows that limα f dµα “ f dµ for functions f P C00 pRν q that are real valued. But then (8) follows.. 328 Download free eBooks at bookboon.com.

(93) Advanced stochastic processes: Part II. Some related results. 2. CONVERGENCE OF POSITIVE MEASURES. 329. (10) ñ (7) Let a ă b be as in (7). Then µα pa, bs can be written in the form ﬀ « ν ź ÿ #Λ p´1q µα p´8, cΛ,j s , (5.49) µα pa, bs “ j“1. ΛĂt1,...,νu. where cΛ,j “ aj , j P Λ, cΛ,j “ bj , j P t1, . . . , νu zΛ. The implication (10) ñ (7) then easily follows from (5.49). The equality in (5.49) can be found in Durrett [46] Theorem 1.1.6 page 7. ś (7) ñ (10) Let a be as in assertion (10). Put F “ νk“1 p´8, ak s. Then the subset F is closed, and since assertion (7) is equivalent to (4) we know that lim supα µα pF q ď µpF q. Since ˆ assertion ˙ ˆ(7)˙is equivalent ˆ to ˙ (5) we know that ˝. lim inf α µα pF q ě lim inf α µα F. ˝. ˝. ě µ F . Since µ F. “ µpF q assertion. (10) follows.. The proof of these implications completes the proof Theorem 5.43.. . Excellent Economics and Business programmes at:. “The perfect start of a successful, international career.” CLICK HERE. to discover why both socially and academically the University of Groningen is one of the best places for a student to be. www.rug.nl/feb/education. 329 Download free eBooks at bookboon.com. Click on the ad to read more.

(94) Advanced stochastic processes: Part II 330 5. SOME RELATED RESULTS. Some related results. 5.44. Remark. The implication (10) ñ (1) in Theorem 5.43 can also be proved by employing the equality ﬀ « ż ż ν ź ν f pxq dµα pxq “ p´1q D1 ¨ ¨ ¨ Dν f pxq µα p´8, xj s dx (5.50) Rν. “ p´1qν. ż. Rν. Rν. ż8 y1. ¨¨¨. ż8 yν. j“1. D1 ¨ ¨ ¨ Dν f pxq dxν . . . dx1 dµα pyq,. where the function f is ν times continuously diﬀerentiable, and where Dj denotes diﬀerentiation with respect to the j-th coordinate, 1 ď j ď ν. The equality in (5.50) can be proved by successive integration. The second equality is a consequence of Fubini’s theorem. 5.45. Definition. A topological space E is called a Polish space if E possesses the following properties: (i) E is separable; (ii) E is metrizable; (iii) There exists a metric d on E that determines the topology and relative to which E is complete. Since E is metrizable property (i) is equivalent with the existence of a countable basis for the topology. 5.46. Lemma. Let E be a Polish space. (a) A closed subset F of E is, with the induced metric, again Polish. (b) An open subset G of E is again Polish. Proof. (a) The proof of assertion (a) is not diﬃcult. If pUj : j P Nq is a countable basis for the topology of E, then pUj X F : j P Nq is a countable basis for the topology on F . Moreover, F is closed and hence it is complete with respect to the induced metric. (b). Let d be a metric on E, which turns E into a complete metric topological space. Then the open subset G is Polish for the metric dG deﬁned by ˇ ˇ ˇ ˇ 1 1 ˇ, ˇ ´ dG px, yq “ dpx, yq ` ˇ (5.51) d px, Gc q d py, Gc q ˇ. where x and y belong to G, where Gc “ EzG and where d px, Gc q “ infc dpx, zq. zPG The separability of G is also clear. The proof of Lemma 5.46 is now complete. 5.47. Theorem. A subset A of a Polish space E is again a Polish space if and only if A is the countable intersection of open subsets of E. Ş Proof. Let A “ jPN Gj , where every Gj is an open subset of E. Let d be a metric on E, which makes E into a Polish space. Deﬁne then the metrics dj. 330 Download free eBooks at bookboon.com.

(95) Advanced stochastic processes: Part II 2. CONVERGENCE OF POSITIVE MEASURES. Some related results 331. on Gj , j P N, as in (5.51) and deﬁne the metric dA on A via 8 ÿ 1 dj px, yq dA px, yq “ . 2j 1 ` dj px, yq j“1. Then, endowed with the relative topology, A is a Polish space with respect to the dA . Conversely, let d ď 1 be a metric on A which is compatible with the topology that A inherits from E, and which turns A into a complete metric space. Let A be the closure of A. ThenŞthere exists a decreasing sequence of open subsets pGn qnPN such that A “ n Gn ; i.e. closed subsets of E are Gδ -subsets. For n P N, n ‰ 0, we deﬁne the open subset An of A as follows: ␣ An “ x P A : there exists an open neighborhood U pxq in E of x * 1 (5.52) for which dpy, zq ă for all z, y P U pxq X A .. n Then the following assertions about the sets An will be proved: (1) For all n P N we have A Ă An . (2) The sets An , nŞP N, are open in A. Ş (3) The inclusion n An Ă A holds, and so by (1) A “ n An .. Since, by (2), the subsets An , n P N, are open in A there exist open subsets On , n P N, of E such that An “ On X A, n P N. It follows that č č č č č A“ On X G m “ An “ On X A “ On X Gn , n. n. n. m. n. and hence, A is a countable intersection of open subsets of E. Next we prove the assertions (1), (2) and (3). (1) Pick x P A, and consider the ball * " 1 . B1{p2nq pxq “ w P A : d pw, xq ă 2n. There exists an open neighborhood U pxq of x in E such that B1{p2nq pxq “ A X U pxq. If y, z belong to A X U pxq we have dpz, yq ď dpz, xq ` dpx, yq ă 1{n. It follows that x P An . This is true for all n P N.. (2) That the subset An is open in A can be seen as follows. Pick x P A. There exists an open neighborhood U pxq in E of x such that dpz, yq ă 1{n for all z, y P A X U pxq. The set U pxq X A is an open neighborhood of x in A. It suﬃces to show that U pxq X A Ă An . To this end choose x1 P U pxq X A. Then U pxq is an open neighborhood in E of x1 as well, and since x1 belongs to A it follows from the deﬁnition of An that x1 is a member of An .. (3) Let x belong to An for all n P N. Then x belongs to A. We will prove that x P A. Let D ď 1 be a metric on E which is compatible with its topology, and which turns E into complete metric space. For the moment ﬁx n P N. Since x P An there exists an open ball Bn in E relative to the metric D centered at x. 331 Download free eBooks at bookboon.com.

(96) Advanced stochastic processes: Part II 332 5. SOME RELATED RESULTS. Some related results. and with radius ă 1{n such that 1 whenever y, z P A X Bn . (5.53) dpz, yq ă n Since x P A there exists xn P A X Bn . In this way we obtain a sequence of balls tBn : n P Nu relative to the metric D centered at x, which we take decreasing, and which are such that the D-radius of Bn is strictly less than 1{n. In addition, we obtain a sequence of points txn : n P Nu in A such that xn P Bn for all n P N. Since the balls Bn are decreasing we have xm P Bn for m ě n. From this fact and (5.53) it follows that d pxm , xn q ă 1{n for m ě n. Consequently, it follows that the sequence txn : n P Nu is a d-Cauchy sequence in A. Since A is d-complete there exists a point x1 P A such d pxn , x1 q ď 1{n, n P N. Since D and d are topologically compatible on A it also follows that limnÑ8 D pxn , x1 q “ 0. We also have limnÑ8 D pxn , xq “ 0, and consequently x “ x1 P A. . This completes the proof of Theorem 5.47.. For a proof of the following theorem the reader is referred to the literature. We will give an outline of a proof. A Gδ -set in a topological space is a countable intersection of open subsets. 5.48. Theorem. A Polish space E is homeomorphic with a Gδ -subset of the Hilbert-cube r0, 1sN , endowed with the product topology.. In the past four years we have drilled. 89,000 km That’s more than twice around the world.. Who are we?. We are the world’s largest oilfield services company1. Working globally—often in remote and challenging locations— we invent, design, engineer, and apply technology to help our customers find and produce oil and gas safely.. Who are we looking for?. Every year, we need thousands of graduates to begin dynamic careers in the following domains: n Engineering, Research and Operations n Geoscience and Petrotechnical n Commercial and Business. What will you be?. careers.slb.com Based on Fortune 500 ranking 2011. Copyright © 2015 Schlumberger. All rights reserved.. 1. 332 Download free eBooks at bookboon.com. Click on the ad to read more.

(97) Advanced stochastic processes: Part II 2. CONVERGENCE OF POSITIVE MEASURES. Some related results 333. Proof of Theorem 5.48. Let d : E ˆ E Ñ r0, 1s be a metric op E which is compatible with its topology, and which turns E into a Polish space, and let pxℓ : ℓ P Nq be a countable dense subset of E. Define the mapping Ψ : E Ñ r0, 1sN by Ψpxq “ pd px, xℓ qℓPN q. Then, as can be checked, the mapping Ψ is a homeomorphism from E onto a subset of r0, 1sN . So far we have not yet used the fact that E, equipped with the metric d is complete; we did use the fact that E is a metrizable separable space. Since E is complete with respect to d and Ψ is a homeomorphism, it follows that the image of E under Ψ, that is A “ Ψ pEq, is complete subspace of the Hilbert cube r0, 1sN . Let D : r0, 1sN ˆ r0, 1sN Ñ r0, 1s be the metric defined by 8 ÿ 2´ℓ |ξℓ ´ ηℓ | , pξℓ qℓPN , pηℓ qℓPN P r0, 1sN . D ppξℓ qℓPN , pηℓ qℓPN q “ ℓ“1. Then D is a metric on r0, 1sN which turns this space into a Polish space. It follows that A is a subset of r0, 1sN which is homeomorphic to a Polish space, and so it itself is Polish. Since it is a Polish subspace of the Polish space r0, 1sN , A is a countable intersection of open subsets of r0, 1sN : see Theorem 5.47. This completes the proof of Theorem 5.48. . The space N of positive integers with the usual metric inherited from the real numbers R is Polish. Then the countable product NN with metric 8 ÿ 1 |mi ´ ni | (5.54) d ptmi u , tni uq “ 2i 1 ` |mi ´ ni | i“1 is Polish. The proofs of Propositions 5.49 and 5.50 are taken from Garrett [57].. 5.49. Proposition. Totally order NN lexicographically. Then every closed subset C of NN has a least element. The lexicographic ordering of NN can be recursively defined. An element a “ pa1 , a2 , . . .q precedes an element b “ pb1 , b2 , . . .q if a1 ď b1 ; however, if a1 “ b1 , then a2 ď b2 ; however, if a1 “ b1 and a2 “ b2 , then a3 ď b3 , and so on. Proof. Let n1 be the least element in N such that there is x “ pn1 , . . .q belonging to C. Let n2 be the least element in N such that there is x “ pn1 , n2 , . . .q belonging to C, and so on. Choosing the ni inductively, let x0 “ pn1 , n2 , n ` 3, . . .q. This x0 satisfies x0 ď x in the lexicographic ordering for every x P C, and x0 belongs to the closure of C in the metric topology introduced in (5.54). This completes the proof of Proposition 5.49. 5.50. Proposition. Let E be a Polish space. Then there exists a continuous surjective mapping F0 : NN Ñ E. Moreover, there exists a measurable function G0 : E Ñ NN such that F0 ˝ G0 pyq “ y for all y P E. Proof. The mapping F0 can be constructed as follows. For a given ε ą 0 there is a countable covering of E by closed sets of diameter less than ε. From this one may contrive a map F from finite sequences tn1 , . . . , nk u in N to closed sets F pn1 , . . . , nk q in E such that. 333 Download free eBooks at bookboon.com.

(98) Advanced stochastic processes: Part II 334. 5. SOME RELATED RESULTS. Some related results. (1) F pHq “ E; Ť (2) F pn1 , . . . , nk q “ 8 ℓ“1 F pn1 , . . . , nk ; ℓq; (3) The diameter of F pn1 , . . . , nk q is less than 2´k .. Then for x “ tni u P NN the sequence Ek “ F pn1 , . . . , nk q is a nested sequence ´k of closed Ş subsets of E with diameters less than 2 , respectively. Thus, the subset k Ek consists ofŞa single point F0 pyq of E. On the other hand, every x P E lies inside some k Ek . Continuity is easy to verify. The mapping G0 can be constructed as follows. The space NN , endowed with the lexicographical ordering is totally ordered, and by Proposition 5.49 every closed subset contains a least element for this order. For y P E the subset F0´1 pyq is closed in NN , and therefore it contains a least element G0 pyq. This assignment is a measurable choice (because it can be performed in countably many steps). Then F0 ˝G0 pyq “ y for y P E. The proof of Proposition 5.50 is complete now. The proof of the following theorem is based on the fact that a Polish space is homeomorphic with a Gδ -subset of the Hilbert cube, which, being a countable product of closed intervals, is a compact metrizable space. 5.51. Theorem. Let µ be a finite positive measure on the Borel field of a Polish space. Then µ is regular in the sense that µpBq “ inf tµpOq : O open, B Ă Ou “ sup tµpKq : K compact, K Ă Bu . (5.55) Proof. Let Ψ and A be as in the proof of Theorem 5.48. Then Ψ : E Ñ A is a homeomorphism. Let µ ě 0 be a finite measure on the Borel field of E. Define the measure ν on the Borel field of r0, 1sN by ‰ “ νpBq “ µ Ψ´1 pB X Aq “ µ rΨ P B X As , B Borel subset of r0, 1sN . (5.56). Then, since the Hilbert cube is compact and complete metrizable, and A is a Gδ -subset of the Hilbert cube, we see that the measure ν is regular on the Borel field of r0, 1sN . It also follows that the restriction of ν to the Borel field of A is regular. However, under the homeomorphism Ψ : E Ñ A the Borel subsets of E are in a one-to-one correspondence with those of A. It easily follows that the measure µ is regular in the sense of (5.55), which completes the proof of Theorem 5.51. 5.52. Theorem. The following assertions hold for Banach spaces. (a) Let E be a separable Banach space. Then its dual unit ball B 1 , endowed with the weak˚ -topology, is a Polish space. (b) (Helly) The set M` ď1 is compact-metrizable, and thus Polish for the vague topology. Let E 1 be the topological dual space of E. The weak˚ -topology is denoted by σ :“ σ pE 1 , Eq. Proof. (a) Let pxn : n P Nq be a sequence in the unit ball B of E of which the linear span is dense in E. Define the mapping Φ : pB 1 , σq Ñ r0, 1sN via the. 334 Download free eBooks at bookboon.com.

(99) Advanced stochastic processes: Part II 2. CONVERGENCE OF POSITIVE MEASURES. Some related results 335. map x1 ÞÑ pă xn , x1 ąq8 n“1 . The mapping Φ is continuous and, by the Theorem of Banach-Alaoglu, the dual unit ball B 1 is compact relative the topology σ pE 1 , Eq. It follows that ΦpB 1 q is a compact subset of r0, 1sN . This image is Polish, and because the inverse of Φ is continuous, B 1 itself is Polish as well. Let r be a positive real number. Let the set M` ďr be defined by ␣ ( ` ν M` ďr “ µ P M : µ pR q ď r ,. and let M` r be given by. ␣ ( ` ν M` r “ µ P M : µ pR q “ r .. ν ˚ (b) Since M` ď1 is a vaguely closed subset of M “ C0 pR q , by the Theorem of Banach-Alaoglu it follows that M` ď1 is compact for the vague topology. The is Polish will be proved in Theorem 5.54. This completes fact that the set M` ď1 the proof of Theorem 5.52. . 5.53. Definition. A subset A of M is a Prohorov subset, if it satisfies the following two conditions: (a) supµPA |µ| pRν q is finite; (b) For every ϵ ą 0 there exists a compact subset K of Rν such that for all measures µ P A.. |µ| pRν zKq ď ϵ. American online LIGS University is currently enrolling in the Interactive Online BBA, MBA, MSc, DBA and PhD programs:. ▶▶ enroll by September 30th, 2014 and ▶▶ save up to 16% on the tuition! ▶▶ pay in 10 installments / 2 years ▶▶ Interactive Online education ▶▶ visit www.ligsuniversity.com to find out more!. Note: LIGS University is not accredited by any nationally recognized accrediting agency listed by the US Secretary of Education. More info here.. 335 Download free eBooks at bookboon.com. Click on the ad to read more.

(100) Advanced stochastic processes: Part II 336 5. SOME RELATED RESULTS. Some related results. ` 5.54. Theorem. (a) The spaces M` ď1 and M1 are Polish with respect to the vague topology. ` (b) The spaces M` ď1 and M1 are Polish with respect to the weak topology.. Proof. The countable collection + # 8 n n ď ÿ ÿ αj δxj : αj P Q, αj ě 0, αj “ 1 n“1. j“1. j“1. is dense in M` 1 for the vague as well as for the weak topology. The countable collection + # 8 n n ď ÿ ÿ αj δxj : αj P Q, α ě 0, 0 ď αj ď 1 n“1. j“1. j“1. is dense in M` ď1 for de vague as well as the weak topology. This can be seen as follows. Let f be a bounded continuous function defined op Rν . Fix ϵ ą 0, and choose a partition of Rν in Borel subsets pAj : j P Nq in such a way that ϵ for all x, y P Aj , and this for all j P N. In addition, |f pxq ´ f pyq| ď µ pRν q choose N P N so large that ¸ ˜ N ÿ µ pAj q }f }8 ď ϵ. µ pRν q ´ j“1. Put aj “ µ pAj q and choose xj P Aj . Then we have ˇ ˇż řN ˇ ˇ a f px q j j ˇ ˇ j“1 µ pRν qˇ ˇ f dµ ´ řN ˇ ˇ j“1 aj ˇ ˇ ˇ ż # +ˇ 8 N ˇ ˇÿ ˇ ˇÿ ν µ pR q ˇ ˇ ˇ ˇ pf pxq ´ f pxj qq dµpxqˇ ` ˇ aj f pxj q 1 ´ řN ďˇ ˇ ˇ ˇj“1 ˇ ˇj“1 Aj j“1 aj ˇ ˇ ż 8 ˇ ˇ ÿ ˇ ˇ f pxqdxˇ `ˇ ˇ ˇj“N `1 Aj + # 8 N ÿ ϵ ÿ µ pAj q ` 2 }f }8 µ pRν q ´ µ pAj q ď 3ϵ. ď µ pRν q j“1 j“1 Appealing another time to the continuity of the function f , and using the fact that the rational numbers are dense in R we obtain the separability of the ` sets M` r and Mďr relative tot the vague as well as the weak topology. We indicate metrics which turn these spaces into Polish spaces. Therefore we choose a sequence of functions pfk : k P Nq in tf P C00 pRν q : 0 ď f ď 1u whose linear span is uniformly dense in C0 pRν q. We also choose a sequence puℓ : ℓ P Nq in C00 pRν q such that 1 ě uℓ`1 ě uℓ ě 0 and such that 1 “ limℓÑ8 uℓ pxq for all x P Rν .. 336 Download free eBooks at bookboon.com.

(101) Advanced stochastic processes: Part II 2. CONVERGENCE OF POSITIVE MEASURES. (a) Define the distance dv on M` ď1 via the formula ˇż ˇ ż 8 ÿ ˇ 1 ˇˇ ˇ, dv pµ, νq “ dµ ´ f dν f j j ˇ j ˇ 2 j“1. Some related results 337. (5.57). ` where µ and ν are members of M` ďr . Supplied with this metric the space Mďr ` is Polish for the vague topology. The spaces Mďr , 0 ă r ď 1, are also closed for the vague topology. Since. M` 1 “ we see that. M` 1. 8 č. n“1. ` M` ď1 zMď1´1{n. (5.58). is a Polish space: see the Theorems 5.47 and 5.52.. (b) Let f0 ” 1 and let the sequences pfk : k P Nq and puℓ : ℓ P Nq be as above. We define the metric dw by the equality ˇż ˇż ˇ ÿ ˇ ż ż 8 ˇ ˇ ˇ ˇ 1 ˇ fj dµ ´ fj dν ˇ , dw pµ, νq “ sup ˇˇ uℓ dµ ´ uℓ dν ˇˇ ` (5.59) ˇ ˇ j 2 ℓPN j“1. ` where µ and ν belong to M` ďr . Supplied with this metric the space Mďr is Polish ` for the weak topology. If we are also able to prove that the space Mďr is complete relative to the metric dw , then it follows that the spaces M` ďr , r ě 0, are Polish. These spaces are also weakly closed. By the equality in (5.58) in Theorem 5.47 then implies that M` 1 is a Polish space as well. Now let pµm : m P Nq be a dw Cauchy sequence in M` ďr . We will prove that this sequence is a Prohorov set in ` Mďr . Choose ϵ ą 0 arbitrary. Then there exists Mϵ P N such that for m and m1 ě Mϵ the inequality ˇż ˇ ż ˇ ˇ ϵ ˇ sup ˇ uℓ dµm ´ uℓ dµm1 ˇˇ ď (5.60) 4 ℓPN. holds. So it follows that. ϵ |µm pRν q ´ µm1 pRν q| ď , 4. m, m1 ě Mϵ .. From (5.60) and (5.61) it follows that ˇ ˇż ż ˇ ϵ ˇ ˇ sup ˇ p1 ´ uℓ q dµm ´ p1 ´ uℓ q dµm1 ˇˇ ď , 2 ℓPN for m and m1 ě Mϵ . Then from (5.62) it follows that ż ż ϵ p1 ´ uℓ q dµm ď p1 ´ uℓ q dµMϵ ` . 2. (5.61). (5.62). (5.63). From (5.63) it then follows that for ℓ ě ℓϵ and m ě Mϵ the following inequality holds: ż ϵ 3ϵ ϵ (5.64) p1 ´ uℓ q dµm ď ` “ . 4 2 4. 337 Download free eBooks at bookboon.com.

(102) Advanced stochastic processes: Part II 338. 5. SOME RELATED RESULTS. Some related results. From (5.64) it then follows that for all ℓ ě Lϵ and for all m P N the inequality ż (5.65) p1 ´ uℓ q dµm ď ϵ.. holds. Then choose the compact subset Kϵ equal the support of the function uLℓ . It follows that µm pRν zKϵ q ď ϵ for all m P N. Let µ be the vague limit of the sequence pµm : m P Nq. This limit exists, because M` ďr is compact for the ν metric dv . Then pick a function u P C00 pR q in such a way that 1 ě u ě 1Kϵ . By the equality ż ż ż ż ż ż f dµm ´ f dµ “ pf ´ f uq dµm ` f udµm ´ f udµ ´ pf ´ f uq dµ we infer the inequality ˇ ˇż ż ˇ ˇ ˇ f dµm ´ f dµˇ ˇ ˇ ˇ ˆż ˙ ˇż ż ż ˇ ˇ ˇ p1 ´ uq dµm ` p1 ´ uq dµ ` ˇ f udµm ´ f udµˇˇ ď }f }8 ˇż ˇ ż ˇ ˇ ν ν ˇ ď }f }8 pµm pR zKϵ q ` µ pR zKϵ qq ` ˇ f udµm ´ f udµˇˇ ˇ ˆ ˙ ˇż ż ˇ ˇ ν ν ď }f }8 µm pR zKϵ q ` sup µm pR zKϵ q ` ˇˇ f udµm ´ f udµˇˇ m ˇ ˇż ż ˇ ˇ (5.66) ď 2ϵ }f }8 ˇˇ f udµm ´ f udµˇˇ .. Since µ “ vague- limm µm from (5.66) it also follows that µ is the weak limit of the sequence pµm : m P Nq.. . The proof Theorem 5.54 is now complete. Part of the proof of Theorem 5.54 comes back in the proof of Theorem 5.55.. 5.55. Theorem. A subset S of M` ďr is relatively weakly compact if and only if S is a Prohorov subset. Proof. First, suppose that S is a Prohorov subset of M` ďr . Let pµm : m P Nq be a sequence in S. We will prove that there exists a subsequence that converges weakly. We may assume that, possibly by passing to a subsequence, that this sequence converges vaguely. By employing the Prohorov property we will show that, in fact, this sequence converges weakly. Let ϵ ą 0 be arbitrary. Then there exists a compact subset K such that µm pK c q ď ϵ and also that µ pK c q ď ϵ. Then choose u P C00 pRν q in such a way that 1 ě u ě 1K . Then we have (see the final part of the proof of Theorem 5.54): ˇż ˇ ż ˇ ˇ ˇ f dµm ´ f dµˇ ˇ ˇ ˇ ˆż ˙ ˇż ż ż ˇ ˇ ˇ p1 ´ uq dµm ` p1 ´ uq dµ ` ˇ f udµm ´ f udµˇˇ ď }f }8 338 Download free eBooks at bookboon.com.

(103) Advanced stochastic processes: Part II 2. CONVERGENCE OF POSITIVE MEASURES. Some related results 339. ˇż ˇ ż ˇ ˇ ď }f }8 pµm pR zKq ` µ pR zKqq ` ˇˇ f udµm ´ f udµˇˇ ˇ ˆ ˙ ˇż ż ˇ ˇ ď }f }8 µm pRν zKq ` sup µm pRν zKq ` ˇˇ f udµm ´ f udµˇˇ m ˇż ˇ ż ˇ ˇ ˇ ď 2ϵ }f }8 ˇ f udµm ´ f udµˇˇ . (5.67) ν. ν. Since µ “ vague- limm µm from (5.67) it also follows that µ is the weak limit of the sequence pµm : m P Nq.. Conversely, suppose that the set S is weakly compact. We will prove that S possesses the Prohorov property. Assume the contrary. Then there exists an η ą 0 and there exists an increasing sequence of compact subsets pKm : m P Nq with the following properties: ˝ Ť (a) Km Ă K m`1 and Rν “ 8 m“1 Km ; c q ě η. (b) For every m P N there exists a measure µm P S such that µm pKm. Then choose a sequence of functions pum P C00 pRν qq such that 1Km ď um`1 ď 1Km`1 . From (b) it follows then that, for m ě ℓ, ż ż ż η ď p1 ´ 1Km q dµm ď p1 ´ 1Kℓ q dµm ď p1 ´ uℓ q dµm . (5.68) Since the subset S is relatively weakly compact, there exists a subsequence pµmk : k P Nq şwhich converges weakly to a measure µ. From (5.68) we then see that η ď p1 ´ uℓ q dµ for all ℓ P N. From this we obtain a contradiction, because the sequence p1 ´ uℓ : ℓ P Nq decreases to 0. This completes the proof of Theorem 5.55.. . 5.56. Corollary. Let pµm : m P Nq be a sequence of measures in M` and let µ also belong to M` . Choose a sequence of functions pfk : k P Nq in tf P C00 pRν q : 0 ď f ď 1u with a linear span that is uniformly dense in C0 pRν q and choose another sequence puℓ : ℓ P Nq in C00 pRν q such that 1 ě uℓ`1 ě uℓ ě 0 and such that 1 “ limℓÑ8 uℓ pxq for all x P Rν . Define the metric dw by the equality ˇż ˇ ÿ ˇż ˇ ż ż 8 ˇ ˇ ˇ 1 ˇˇ ˇ ˇ dw pν1 , ν2 q “ sup ˇ uℓ dν2 ´ uℓ dν1 ˇ ` fj dν2 ´ fj dν1 ˇˇ , ˇ j 2 ℓPN j“1 where ν1 and ν2 belong to M` . Then the following assertions are equivalent: (i) The sequence pµm : m P Nq converges weakly to µ; (ii) limmÑ8 dw pµm , µq “ 0. Proof. The proof is left as an exercise for the reader.. 339 Download free eBooks at bookboon.com. .

(104) Advanced stochastic processes: Part II 340 5. SOME RELATED RESULTS. Some related results. 3. A taste of ergodic theory In this section we will formulate and prove the pointwise ergodic theorem of Birkhoﬀ. We also indicate its relation with the strong law of large numbers. We will also show that the strong law of large numbers (SLLN) implies the weak law of large numbers (WLLN). However, we begin with von Neumann’a ergodic theorem in a Hilbert space. In what follows the symbol H stands for a (complex) Hilbert space with inner-product ⟨¨, ¨⟩ and norm }¨}. An operator U : H Ñ H is called unitary if it satisﬁes U ˚ U “ U U ˚ “ I. An operator P : H Ñ H is called an orthogonal projection if P ˚ “ P “ P 2 . Let L be closed subspace of H. Then H can be written as H “ L ‘ LK “ P H ` pI ´ P q H,. where P : H Ñ H is an orthogonal projection with range L. The following theorem is the same as Theorem 7.1 in Romik [115].. .. 340 Download free eBooks at bookboon.com. Click on the ad to read more.

(105) Advanced stochastic processes: II 3. APart TASTE OF ERGODIC THEORY. Some related results 341. 5.57. Theorem. Let H be a Hilbert space, and let U be a unitary operator on H. Let P be the orthogonal projection operator onto the subspace N pU ´ Iq (the subspace of H consisting of U -invariant vectors). For any vector v P H the equality n´1 1 ÿ k U v “ P v. lim nÑ8 n k“0 + # n´1 ÿ 1 U k : n P N converges holds. (Equivalently, the sequence of operators n k“0 to P in the strong operator topology.) Proof. Define the subspace V Ă H by V “ N pU ´Iq “ tv P H : U v “ vu “ N pU ˚ ´ Iq. Then V “ pRpU ´ IqqK :“ tv P H : ⟨pU ´ Iq u, v⟩ “ 0 for all u P Hu ,. (5.69). where RpU ´ Iq is the range of the operator U ´ I, i.e. RpU ´ Iq “ tU u ´ u : u P Hu . Let L be any linear subspace of H. From Hilbert space techniques it is known ` ˘K that LK coincides with the closure of the linear subspace L. From these observations it follows that the subspace N pU ´ Iq ` R pU ´ Iq is dense in H, and that the subspaces N pU ´ Iq and R pU ´ Iq are orthogonal. Moreover, we have › › n´1 › › 1 n´1 › ÿ k › 1 ÿ ›› k ›› U v› ď (5.70) U v ď }v} , v P H. › › n k“0 › n k“0. Define the subspace L Ă H by # L“. v P H : lim. nÑ8. n´1 ÿ. +. 1 U kv “ P v , n k“0. (5.71). where P is the orthogonal projection onto the space V “ N pU ´Iq. From (5.70) it follows that L is a closed subspace of H. If v P V , i.e. if U v “ v, then v belongs to L. If v “ pU ´ Iq u belongs to the range of U ´ I, then n´1 n´1 1 ÿ k 1 1 ÿ k U v“ U pU ´ Iq u “ pU n ´ Iq u, n k“0 n k“0 n. (5.72). and so by (5.70) and (5.71) it follows that n´1 1 ÿ k lim U v “ 0 “ P v, nÑ8 n k“0. (5.73). whenever v belongs to RpU ´ Iq. Again appealing to (5.70) shows that (5.73) also holds for v belonging to the closure of RpU ´ Iq. Altogether it shows that the subspace L coincides with H. This completes the proof of Theorem 5.57. . 341 Download free eBooks at bookboon.com.

(106) Advanced stochastic processes: Part II 342 5. SOME RELATED RESULTS. Some related results. Let pΩ, F, P; T q be a measure preserving system. We associate with the measure preserving mapping T : Ω Ñ Ω an operator UT on the Hilbert space L2 pΩ, F, Pq deﬁned by UT pf q “ f ˝T , f P L2 pΩ, F, Pq. The fact that T is measure preserving implies that UT˚ UT “ I: “ ‰ “ ‰ ⟨UT f, UT g⟩ “ E UT f UT g “ E f ˝ T g ˝ T “ E rpf gq ˝ T s “ E rf gs “ ⟨f, g⟩ . (5.74) From (5.74) it follows that UT˚ UT “ I. In order that UT is a unitary operator it should also be surjective. Since the range of UT is closed, this is the case provided that the set of functions f ˝ T , f P L2 pΩ, F, Pq, constitutes a dense subspace of L2 pΩ, F, Pq. The latter is true if the mapping T has the property that there exists a measurable mapping Tr : Ω Ñ Ω such that Tr ˝ T pωq “ ω for r deﬁned by U r f “ f ˝ Tr, f P L2 pΩ, F, Pq P-almost all ω. Then the operator U is the adjoint of UT . This can be seen as follows. Now we do not only have rf “ U r f ˝ T “ f ˝ Tr ˝ T “ f , f P L2 pΩ, F, Pq. UT˚ UT “ I, but we also have UT U Hence, we see ´ ¯ r “ UT˚ UT U r “ UT˚ . r “ pUT˚ UT q U U. Note also that the subspace N pU ´ Iq consists exactly of the invariant (squareintegrable) random variables, or equivalently those random variables which are measurable with respect to the σ-algebra I of invariant events. Recalling the discussion of conditional expectations in Theorem 1.4, item (11), in Chapter 1, we also see that the orthogonal “ ˇ ‰ projection operator P is exactly the conditional expectation operator E ¨ ˇ I with respect to the σ-algebra of invariant events I. Thus, Theorem 5.57 applied to this setting gives the following result. 5.58. Theorem (The L2 ergodic theorem). Let pΩ, F, P; T q be a measure preserving system. For any random variable X P L2 pΩ, F, Pq the equality n´1 “ ˇ ‰ 1 ÿ lim X ˝ Tk “ E X ˇ I nÑ8 n k“0. (5.75). holds in L2 pΩ, F, Pq. In particular, if the system is ergodic then n´1 1 ÿ X ˝ T k “ E rXs . nÑ8 n k“0. L2 - lim. (5.76). Since the operator S : L2 pΩ, F, Pq Ñ L2 pΩ, F, Pq, deﬁned by Sf “ f ˝ T , f P L2 pΩ, F, Pq, is not necessarily unitary, Theorem 5.58 requires a proof. It only satisﬁes S ˚ S “ I. The proof below is based on the proof of Theorem 5.66 below. Proof. Theorem 5.58 is a consequence of Theorem 5.59 which includes the L -version of Theorem 5.58. More precisely, we have to prove that »ˇ ˇ2 fi ˇ 1 n´1 “ ˇ ‰ˇˇ ˇ ÿ (5.77) X ˝ T k ´ E X ˇ I ˇ fl “ 0. lim E –ˇ nÑ8 ˇ n k“0 ˇ 1. 342 Download free eBooks at bookboon.com.

(107) Advanced stochastic processes: Part II. Some related results. 3. A TASTE OF ERGODIC THEORY. 343. Let X P L2 pΩ, F, Pq be bounded. Then we can use Theorem 5.59 together Lebesgue’s theorem of dominated convergence that the equality in (5.77) holds for X. A general function Xş P L2 pΩ, F, Pqş can be approximated by bounded functions in L2 -sense. Since |f ˝ T |2 dµ “ |f |2 dµ, and so the convergence in (5.77) also holds for all L2 -functions. The precise argument follows as in (5.122) below with Sf “ f ˝ T , f P L2 pΩ, F, Pq, and relative to the L2 -norm instead of the L1 -norm. So let f belong to L2 pΩ, F, Pq, and let M ą 0 be an arbitrary real ›number. Then we› have: › › 1 n´1 › › ÿ k S f ´ Pµ f › › › 2 › n k“0 L ›˜ ›˜ › › ¸ ¸ › 1 n´1 › 1 n´1 › › ÿ ÿ ` ` ˘ ˘ › › › › ď› S k ´ Pµ f 1t|f |ăM u › ` › S k ´ Pµ f 1t|f |ěM u › › n k“0 › 2 › n k“0 › 2 L L ¨ ˇ˜ ˛1{2 ˇ ¸ 2 ˆż ˙1{2 ż ˇ n´1 ` ˘ˇˇ ˇ 1 ÿ k 2 ˝ ‚ ď S ´ Pµ f 1t|f |ăM u ˇ dP `2 |f | dP . ˇ ˇ n k“0 ˇ t|f |ěM u (5.78). As in the proof of Theorem 5.66 in (5.78) we ﬁrst let n Ñ 8, and then M Ñ 8 to obtain the L2 -convergence in Theorem 5.58.. Join the best at the Maastricht University School of Business and Economics!. Top master’s programmes • 3 3rd place Financial Times worldwide ranking: MSc International Business • 1st place: MSc International Business • 1st place: MSc Financial Economics • 2nd place: MSc Management of Learning • 2nd place: MSc Economics • 2nd place: MSc Econometrics and Operations Research • 2nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier ‘Beste Studies’ ranking 2012; Financial Times Global Masters in Management ranking 2012. Maastricht University is the best specialist university in the Netherlands (Elsevier). Visit us and find out why we are the best! Master’s Open Day: 22 February 2014. www.mastersopenday.nl. 343 Download free eBooks at bookboon.com. Click on the ad to read more.

(108) Advanced stochastic processes: Part II 344 5. SOME RELATED RESULTS. Some related results. The pointwise ergodic theorem of Birkhoﬀ requires some more work. In what follows pΩ, F, µq is positive measure space, and T : Ω Ñ Ω is a measure preserving mapping, i.e. µ tT P Au “ µ tT ´1 Au “ µ tAu for all A P F with µ tAu ă 8. 1 An ş equivalent ş formulation reads as follows. For all f P L pΩ, F, µq the equality f ˝ T dµ “ f dµ holds. In other words the quadruple pΩ, F, P; T q is a measure preserving system, or dynamical system. The operator Pµ in (5.80) is a projection mapping from L1 pΩ, F, µq onto a space consisting of T -invariant functions. Hence a function of the form g “ Pµ f satisﬁes g ˝ T “ g µ-almost everywhere, and so g is measurable with respect to the invariant σ-ﬁeld I. In addition, if h is a bounded, T -invariant function in L1 pΩ, F, µq, then we have ż ż ż (5.79) pPµ f q h dµ “ Pµ pf hq dµ “ f h dµ. In other words the function Pµ f is the µ-conditional expectation of the function f on the σ-ﬁeld of invariant subsets. A measure preserving system pΩ, F, µ; T q is called ergodic if a T -invariant function is constant µ-almost every, and so the σ-ﬁeld I is trivial, i.e. I consists, up to sets of µ-measure zero, of the void set and of Ω. 5.59. Theorem (The pointwise ergodic theorem in L1 ). Let pΩ, F, µ; T q be a measure preserving system. For any function f P L1 pΩ, F, µq the equality n´1 1 ÿ f ˝ T k “ Pµ f nÑ8 n k“0. lim. (5.80). holds µ-almost everywhere. In particular, if the system is ergodic then the equality ż n´1 1 ÿ k lim f ˝ T “ f dµ. (5.81) nÑ8 n k“0 holds µ-almost everywhere. If µ is a probability measure, “ ˇ ‰then the limits in 1 (5.80) and (5.81) also hold in L -sense, and Pµ f “ Eµ f ˇ I .. Proof. The proof of Theorem 5.59 follows from Theorem 5.66 and its Corol lary 5.67 with µ pΩq “ 1, and Sf “ f ˝ T , f P L1 pΩ, F, µq. Let pΩ0 , F0 , Pq be a probability space, and let Xj : Ω0 Ñ R, j “ 0, 1, . . ., be a sequence of independent and identically distributed variables (i.i.d.). řn´1Let us show that the SLLN is a consequence: see Theorem 2.54. Put Sn “ k“0 Xk . 5.60. Theorem (Strong law of large numbers). The equality. Sn “ α, holds P-almost surely nÑ8 n for some finite constant α, if and only if E r|Xk |s ă 8, and then α “ E rX1 s. lim. Proof. Let Ω “ RN , endowed with the product σ-ﬁeld F “ b8 j“0 Bj where Bj is the Borel ﬁeld on R. Deﬁne the probability measure µ on F by µpAq “ E r1A pX0 , X1 , . . .qs “ P rpX0 , X1 , . . .q P As , A P F.. 344 Download free eBooks at bookboon.com.

(109) Advanced stochastic processes: Part II 3. A TASTE OF ERGODIC THEORY. Some related results 345. Put Sf px0ş, x1 , . . .q “ f px1 , x2 , . . .q, f P L1 pΩ, F, µq, px0 , x1 , . . .q P Ω. Then ş Sf dµ “ f dµ, f P L1 pΩ, F, µq. The assertion in Theorem 5.60 follows from Theorem 5.66 and its Corollary 5.67 by applying them to the function f0 : Ω Ñ R deﬁned by f0 px0 , x1 , . . .q “ x0 , px0 , x1 , . . .q P Ω. Then n´1 ÿ k“0. n´1 n´1 ÿ ÿ ˘ f0 pXk , Xk`1 , . . .q “ Xk , S k f0 pX0 , X1 , X2 , . . .q “. `. k“0. k “ 0, 1, . . . ,. k“0. and hence Theorem 5.60 is a consequence of Theorem 5.66 and its Corollary 5.67. Theorem 5.60 also follows from Theorem 5.59 by applying it to the mapping T : Ω Ñ Ω given by T px0 , x1 , . . .q “ px1 , x2 , . . .q, px0 , x1 , . . .q P Ω. We will formulate some of the results in terms of positivity preserving operators S : L1 pΩ, F, µq Ñ L1 pΩ, F, µq. 5.61. Lemma. Let S : L1 pΩ, F, µq Ñ L1 pΩ, F, µq be a linear map, and let f ě 0 belong to L1 pΩ, F, µq. Then the following assertions are equivalent: (i) min pSf, 1q “ S pmin pf, 1qq; (ii) max pSf ´ 1, 0q “ S pmax pf ´ 1, 0qq; (iii) min pSf, 1q ď S pmin pf, 1qq, and max pSf ´ 1, 0q “ S pmax pf ´ 1, 0qq. Suppose that for every f ě 0, f P L1 pΩ, F, µq the operator S satisfies one, and hence all of the conditions (i), (ii) and (iii). In addition, assume that ż ż (5.82) |Sf | dµ ď f dµ, for all f P L1 pΩ, F, µq, f ě 0.. 1 Then S is positivity preserving in the sense that ş f ě 0, f P şL pΩ, F, µq, implies Sf ě 0, and contractive in the sense that |Sf | dµ ď |f | dµ for all f P L1 pΩ, F, µq. Then the equivalent conditions (iv), (v), (vi). and (vii) given by. (iv) The equality S pf gqŞ“ pSf q pSgq holds for all f P L1 pΩ, F, µq, and for all g P L1 pΩ, F, µq L8 pΩ, F, µq, (v) S pmin pf, gqq “ min pSf, Sgq is true for all f , g P L1 pΩ, F, µq, (vi) The equality S pmax pf ´ 1, 0qq “ max pSf ´ 1, 0q holds for every f P L1 pΩ, F, µq. (vii) The S1tf ą1u “ 1tSf ą1u holds for every f P L1 pΩ, F, µq. are also true. Moreover, (vi) implies that if the assertions (i), (ii) and (iii) are true for all positive functions f in L1 pΩ, F, µq, then they are true for all functions f in L1 pΩ, F, µq. If the measure µ is finite, then all assertions (i), (ii), (iii) (for all f ě 0, f P L1 pΩ, F, µq,) and (iv), (v), (vi) and (vii) are equivalent. Finally, the operator S : L1 pΩ, F, µq Ñ L1 pΩ, F, µq is continuous, more precisely, ż ż ż (5.83) |Sf | dµ “ S |f | dµ ď |f | dµ, f P L1 pΩ, F, µq .. 5.62. Remark. Assertion (iv) also holds for all f, g P L1 pΩ, F, µq The equality in (v) can be replaced with. Ş. S pmax pf, gqq “ max pSf, Sgq for all f, g P L1 pΩ, F, µq.. 345 Download free eBooks at bookboon.com. L2 pΩ, F, µq.. (5.84).

(110) Advanced stochastic processes: Part II 346 5. SOME RELATED RESULTS. Some related results. The latter is true because minpf, gq ` maxpf, gq “ f ` g. Proof of Lemma 5.61. The equivalence of the assertions (i), (ii) and (iii) follows from the following identities: min pSf, 1q ` S pmax pf ´ 1, 0qq “ Sf “ min pSf, 1q ` max pSf ´ 1, 0q .. Now assume that for all f ě 0, the operator S satisﬁes (i), (ii) or (iii), and assume that S is contractive in the sense of (5.82). Let f ě 0 belong to L1 pΩ, F, µq, and put fn “ max pf ´ n´1 , 0q. ş Then the sequence pf ´ fn qn decreases to 0, and hence, by (5.82), limnÑ8 |Sf ´ Sfn | dµ “ 0. Then there exists a subsequence pfnk qk such that the sequence pSfnk qk converges to Sf µalmost everywhere. Hence Sf ě 0 µ-almost everywhere. In fact it follows that the sequence pSfn qn increases to Sf µ-almost everywhere. Let f P L1 pΩ, F, µq, and write f “ f` ´ f´ where f` “ max pf, 0q, and f´ “ max p´f, 0q. Then |f | “ f` ´ f´ , and |Sf | ď Sf` ` Sf´ . From (5.82) it the follows that ż ż ż ż ż |Sf | dµ ď pSf` ` Sf´ q dµ “ S pf` ` f´ q dµ “ S |f | dµ ď |f | dµ. (5.85). > Apply now redefine your future. - © Photononstop. AxA globAl grAduAte progrAm 2015. axa_ad_grad_prog_170x115.indd 1. 19/12/13 16:36. 346 Download free eBooks at bookboon.com. Click on the ad to read more.

(111) Advanced stochastic processes: Part II 3. A TASTE OF ERGODIC THEORY. Some related results 347. The inequalities in (5.85) prove the contraction property of the operator S. Next we prove the assertions (iv), (v) and (vi) starting from (i), (ii) or Ş (iii) for all f P 1 1 L pΩ, F, µq, f ě 0. Let the function f ě 0 belong to L pΩ, F, µq L2 pΩ, F, µq. Then we write ż8 n2 ÿn ` ˘ 2 ´n`1 max pf ´ α, 0q dα “ sup 2 max f ´ j2´n , 0 , f “2 n. 0. ad so. ` ˘ S f2 “ 2. ż8 0. j“1. S max pf ´ α, 0q dα. “ sup 2. ´n`1. “ sup 2. ´n`1. n. n2 ÿn. ` ` ˘˘ S max f ´ j2´n , 0. n2 ÿn. ` ˘ max Sf ´ j2´n , 0. j“1. (apply assertion (ii)). n. “2. ż8 0. j“1. max pSf ´ α, 0q dα “ pSf q2 .. (5.86). The equality in (5.86) shows that the assertion in (iv) is true provided that f “ g ě 0. For general f “ g we split f in Ş its positive and negative part. For general f and g belonging to L1 pΩ, F, µq L2 pΩ, F, µq we write 2f g “ g 2 . Altogether this shows assertion (iv). (iv) ñ (v) Let f belong pf ` gq2 ´ f 2 ´Ş to L1 pΩ, F, µq L2 pΩ, F, µq. Then we write ż 2 8 f2 |f | “ dt, π 0 t2 ` f 2 and so by assertion (iv) we get. 2 S |f | “ π. ż8 0. S. ". f2 t2 ` f 2. *. dt. (for explanation see below: equality (5.89)) ż 2 8 S pf 2 q “ dt π 0 t2 ` S pf 2 q ((iv) implies S pf 2 q “ pSf q2 ) 2 “ π. ż8 0. pSf q2 dt “ |Sf | . t2 ` pSf q2. (5.87). The equality in (5.87) shows that (5.84) is true for g “ ´f . For general Ş f, g P L1 pΩ, F, µq L2 pΩ, F, µq we write 2 max pf,Ş gq “ |f ´ g| ` f ` g. Conse1 quently, assertion (v) follows for f, g P L pΩ, F, µq L2 pΩ, F, µq. If f and g are arbitrary functions in L1 pΩ, F, µq, then we approximate them by fn :“ f 1t|f |ďnu. 347 Download free eBooks at bookboon.com.

(112) Advanced stochastic processes: Part II 348 5. SOME RELATED RESULTS. Some related results. and by gn :“ g1t|g|ďnu respectively. This shows that assertion (v) is a consequence of (iv), except that the proof of the second equality in (5.87) is not provided yet. In order to prove this equality it suﬃces to prove that, for a ą 0 and g ě 0, g P L1 pΩ, F, µq the equality " * Sg g S “ (5.88) a`g a ` Sg. holds. By assertion (iv) we have " * " * " * g ag g S pa ` Sgq “ S `S Sg a`g a`g a`g * " 2 * " g ag `S “ Sg. “S a`g a`g. (5.89). The equality in (5.89) shows the validity of (5.88). Therefore the second equality in (5.87) is proved now.. (ii) plus (v) ñ (vi) We apply (5.84), which is equivalent to (v), with f P L1 pΩ, F, µq arbitrary and g “ 0 to obtain S pmax pf ´ 1, 0qq “ S pmax pmax pf, 0q ´ 1, 0qq. (employ assertion (ii)) “ max pS max pf, 0q ´ 1, 0q (apply assertion (v)) “ max pmax pSf, S0q ´ 1, 0q. “ max pmax pSf, 0q ´ 1, 0q “ max pSf ´ 1, 0q .. Hence, assertion (vi) follows from (ii) and (v).. (vi) ñ (v) Let f P L1 pΩ, F, µq. By assertion (vi) we have. S max pf, 0q “ lim S max pf ´ ε, 0q “ lim max pSf ´ ε, 0q “ max pSf, 0q . εÓ0. εÓ0. (5.90) Whence, S max pf, 0q “ max pSf, 0q. Since |f | “ 2 max pf, 0q ´ f we easily infer S |f | “ |Sf |, and assertion (v) follows: see the proof of the implication (iv) ñ (v). Ş (ii) plus (v) ñ (iv) Let f belong to L1 pΩ, F, µq L2 pΩ, F, µq. Then we write ż8 2 max p|f | ´ α, 0q dα, f “2 0. and so by assertion (ii) and (v) we get ż8 ` 2˘ S f “2 max pS |f | ´ α, 0q dα 0 ż8 “2 max p|Sf | ´ α, 0q dα 0. “ |Sf |2 “ pSf q2 ,. 348 Download free eBooks at bookboon.com.

(113) Advanced stochastic processes: Part II. Some related results. 3. A TASTE OF ERGODIC THEORY. 349. and hence (iv) follows. (vi) ñ (vii) Let f belong to L1 pΩ, F, µq. Then 1tf ą1u is µ-integrable as well. Then we have S1tf ą1u “ lim S pmin pm max pf ´ 1, 0qqq mÑ8. “ lim pmin pm max pSf ´ 1, 0qqq “ 1tSf ą1u . mÑ8. (5.91). The equality of the ultimate terms in (5.91) proves the implication (vi) ñ (vii).. (vii) ñ (vi) Let f belong to L1 pΩ, F, µq. Then the functions 1tf ąαu , α ą 0, are µ-integrable as well. We have ż8 ż8 1tf ´1ąαu dα “ S1tf ´1ąαu dα S pmax pf ´ 1, 0qq “ S 0 0 ż8 “ 1tSf ´1ąαu dα “ max pSf ´ 1, 0q . (5.92) 0. The equality of the ultimate terms in (5.92) proves the implication (vii) ñ (vi).. If the measure µ is ﬁnite, then the constant functions belong to L1 pΩ, F, µq. Since S1 “ 1, it is easy to see that assertion (iv) implies assertion (i), and hence by what is proved above, we see that for a ﬁnite measure µ all assertion (i) through (vi) are equivalent. The equality and inequality in (5.83) follow from assertion (v) and the inequality in (5.82). This completes the proof of Lemma 5.61. . 349 Download free eBooks at bookboon.com. Click on the ad to read more.

(114) Advanced stochastic processes: Part II 350 5. SOME RELATED RESULTS. Some related results. 5.63. Proposition. Let g and h be functions in L1 pΩ, F, µq. Define, for ´8 ă tg,hu a ă b ă 8, the subset Ca,b by tg,hu. Ca,b. “ tg ă a ă b ă hu .. (5.93). Then the following equality holds:. S1C tg,hu “ 1C tSg,Shu . a,b. (5.94). a,b. t´h,´gu. Proof. Since C´b,´a “ tg ă a ă b ă hu we may assume that b ą 0. If a ă 0, then 1tgăaăbăhu “ 1t´gą´au 1thąbu . From assertions (iv) and (vii) of Lemma 5.61 it follows that S1tgăaăbăhu “ 1t´Sgą´au 1tShąbu “ 1tSgăaăbăShu ,. and consequently (5.94) follows for a ă 0 ă b. If a “ 0, then we replace a with a ´ ε and let ε Ó 0. If a ą 0, then we consider, for 0 ă ε ă a, 1tgďa´εăbăhu “ 1thąbu ´ 1tgąa´εu 1thąbu .. (5.95). Another application of the assertions (iv) and (vii) of Lemma 5.61 then yields by employing (5.95) the equality: S1tgďa´εăbăhu “ 1tShąbu ´ 1tSgąa´εu 1tShąbu “ 1tSgďa´εăbăShu .. (5.96). In (5.96) we let ε Ó 0 to obtain the equality in (5.93) for 0 ă a ă b. This completes the proof of Proposition 5.63. We also need the following proposition. It will be used with gn “ hn of the form n´1 1 ÿ k S f where f P L1 pΩ, F, µq. hn “ n k“0. 5.64. Proposition. Let tgn un and thn un be sequences in L1 pΩ, F, µq with the property that for every c ą 0 the subsets tsupn |gn | ą cu and tsupn |hn | ą cu have tg u ,th u finite µ-measure. Define, for ´8 ă a ă b ă 8, the subset Ca,bn n n n by * " tgn un ,thn un (5.97) “ lim inf gn ă a ă b ă lim sup hn . Ca,b nÑ8. nÑ8. Then the following equality holds:. S1C tgn un ,thn un “ 1C tSgn u,tShn un . a,b. (5.98). a,b. Proof. We write the function 1C tgn un ,thn un as follows: a,b. 1C tgn un ,thn un “ sup inf1 sup inf1 a,b. min. max. 1 1 N1 N1 N2 N2 N1 ďn1 ďN1 N2 ďn2 ďN2. 1tgn. 1 ăaăbăhn2. u,. (5.99). where the suprema and inﬁma are monotone limit operations in L1 pΩ, F, µq. An appeal to assertion (v) in Lemma 5.61, to (5.83), and to (5.84) the equality in (5.99) implies S1C tgn un ,thn un “ sup inf1 sup inf1 a,b. min. max. 1 1 N1 N1 N2 N2 N1 ďn1 ďN1 N2 ďn2 ďN2. S1tgn ăaăbăhn u . 1 2. 350 Download free eBooks at bookboon.com. (5.100).

(115) Advanced stochastic processes: II 3. APart TASTE OF ERGODIC THEORY. Some related results 351. The equality in (5.96) in combination with (5.100) then shows S1C tgn un ,thn un “ sup inf1 sup inf1. min. max. 1 1 N1 N1 N2 N2 N1 ďn1 ďN1 N2 ďn2 ďN2. a,b. “ 1C tSgn un ,tShn un .. 1tSgn. 1 ăaăbăShn2. u (5.101). a,b. . The equality in (5.101) completes the proof of Proposition 5.64.. 5.65. Theorem (Maximal ergodic theorem). Let S : L1 pΩ, F, µq Ñ L1 pΩ, F, µq be a linear map, which is positivity preserving and contractive. So that f P L1 pΩ, F, µq and f ě 0 implies Sf ě 0 and }Sf }1 ď }f }1 . Define for f P L1 pΩ, F, µq the to S corresponding maximal function fr by n´1 1 ÿ k fr “ sup S f. nPN n k“0. (5.102). Then the following assertions are valid:. (a) If f belongs to L pΩ, F, µq, then. ż. f dµ ě 0; tfrą0u (b) If, in addition, min pSf, 1q “ S pminpf, 1qq, for all f P L1 pΩ, F, µq, f ě 0, then for any a ą 0 and any f P L1 pΩ, F, µq, the following inequalities hold: ! ) r µ tSf ą au ď µ tf ą au , and aµ f ą a ď }f }1 . (5.103) 1. Observe that the second inequality in (5.103) resembles the Doob’s maximal inequality for sub-martingales: see Theorem 5.110 or Proposition 3.107. Proof. (a) Let f P L1 pΩ, F, µq, and deﬁne, for n a positive integer, the function hn by ˜ ¸ k ÿ hn “ max max 0, Sj f . (5.104) 0ďkďn´1. j“0. Then we have hn`1 ě hn ě 0, and for ω P Ω such that hn`1 pωq ą 0, we have Shn pωq`f pωq ěn`1 pωq. The latter inequality is a consequence of the inequality ˜ ˜ ¸¸ k ÿ max max 0, Sj f f ` Shn “ f ` S 0ďkďn´1. ě f ` max S 0ďkďn´1. ě max. 0ďkďn´1. “ max. 0ďkďn´1. ˜. ˜. ˜. max 0,. ˜. j“0 k ÿ. Sj f. j“0. max f ` S0, f `. ˜. ˜. max f,. k`1 ÿ j“0. Sj f. k ÿ. ¸¸ S. j`1. f. j“0. ¸¸. “ max. ¸¸. 0ďkďn. ˜. k ÿ. j“0. 351 Download free eBooks at bookboon.com. Sj f. ¸. .. (5.105).

(116) Advanced stochastic processes: Part II 352 5. SOME RELATED RESULTS. Some related results. From (5.105) it readily follows that f ` Shn ě hn`1 on thn`1 ą 0u .. (5.106). Notice that in the arguments leading to hn`1 ě hn , and also to (5.105), we employed the fact that g ě 0, g P L1 pΩ, F, µq, implies Sg ě 0. From (5.106) we infer ż ż thn`1 ą0u. f dµ ě. phn`1 ´ Shn q dµ ż ż “ hn`1 dµ ´ Shn dµ thn`1 ą0u thn`1 ą0u ż ż ż ż ě hn`1 dµ ´ Shn dµ ě hn`1 dµ ´ hn dµ ż “ phn`1 ´ hn q dµ ě 0. (5.107) thn`1 ą0u. From (5.107) we obtain ż ż ż f dµ “ lim f dµ ě 0. f dµ “ Ť nÑ8 th 8 tfrą0u n`1 ą0u n“0 thn`1 ą0u The inequality in (5.108) entails assertion (a).. (5.108). Need help with your dissertation? Get in-depth feedback & advice from experts in your topic area. Find out what you can do to improve the quality of your dissertation!. Get Help Now. Go to www.helpmyassignment.co.uk for more info. 352 Download free eBooks at bookboon.com. Click on the ad to read more.

(117) Advanced stochastic processes: II 3. APart TASTE OF ERGODIC THEORY. Some related results 353. (b) Let f ě 0 belong to L1 pΩ, F, µq, and ﬁx a ą 0. We ﬁrst prove that µ tSf ą au ď µ tf ą au, i.e. the ﬁrst inequality in (5.103). Let m be a positive integer. By the extra hypothesis in (b), together with assertion (ii) in Lemma 5.61, we have min pm max pSf ´ a, 0q , 1q “ S pmin pm max pf ´ a, 0q , 1qq .. (5.109). From (5.109) we deduce ż ż min pm max pSf ´ a, 0q , 1q dµ “ S pmin pm max pf ´ a, 0q , 1qq dµ ż ď min pm max pf ´ a, 0q , 1q dµ. (5.110) In (5.110) we let m tend to 8 to obtain: ż µ tSf ą au “ lim min pm max pSf ´ a, 0q , 1q dµ mÑ8 ż ď lim min pm max pf ´ a, 0q , 1q dµ “ µ tf ą au . mÑ8. (5.111). This proves the ﬁrst inequality in (5.103). In order to show the second inequality in (5.103) we proceed as follows. Let f be a member of L1 pΩ, F, µq, and deﬁne, ! ) always for a ą 0 ﬁxed, the subset D by D “ fr ą a . Here fr is as in (5.106). In addition, deﬁne for n a positive integer, the subset Dn by n ď ( ␣ k Dn “ S f ą a X D. k“0. Then we have n n ÿ ␣ ( ÿ n`1 µ tDn u ď }f }1 , µ Skf ą a ď µ tf ą au “ pn ` 1qµ tf ą au ď a k“0 k“0 and so µ tDn u is ﬁnite. We also have D Ą Dn`1 Ą Dn , and D “ because for f P L1 pΩ, F, µq we have it follows that aµ tDn u “ “. ż. tfrąau ż. 8 ď. Dn . Hence,. n“1. Č fr ´ a ď fr ´ a1Ą Dn ď pf ´ a1Dn q,. a1Dn dµ “. Č tpf ´a1 Dn qą0u. ż. tfr´aą0u. a1Dn dµ ď. pa1Dn ´ f q dµ `. ż. ż. Č tpf ´a1 Dn qą0u. Č tpf ´a1 Dn qą0u. f dµ. (apply assertion (a) to the ﬁrst terem in (5.112)) ż ď0` f dµ ď }f }1 . Č tpf ´a1 Dn qą0u 353. Download free eBooks at bookboon.com. a1Dn dµ (5.112). (5.113).

(118) Advanced stochastic processes: Part II 354. 5. SOME RELATED RESULTS. Some related results. ! ) In (5.113) we let n tend to 8 and infer aµ fr ą a ď }f }1 . This is the second inequality in (5.103), and completes the proof of Theorem 5.65. . 5.66. Theorem (Theorem of Birkhoﬀ). Let S : L1 pΩ, F, µq Ñ L1 pΩ, F, µq be a linear operator such that for every f ě 0, f P L1 pΩ, F, µq, the following two conditions are satisfied: (i) S pmin pf,ş1qq “ min pSf, ş 1q; (ii) }Sf }1 :“ |Sf | dµ ď f dµ “ }f }1 .. (It follows that all properties mentioned in Lemma 5.61 are available as well as the Propositions 5.63 and 5.64, and Theorem 5.65.) Then for every f P L1 pΩ, F, µq the pointwise limit n´1 1 ÿ k S f “: Pµ f nÑ8 n k“0. lim. (5.114). exists µ-almost everywhere. In addition, Pµ f belongs to L1 pΩ, F, µq, and the operator Pµ is a projection operator, i.e. Pµ2 “ Pµ , with the following properties: ş ş (a) |Pµ f | dµ ď |f | dµ, and (b) SPµ f “ Pµ Sf “ Pµ f , where f belongs to L1 pΩ, F, µq. If the measure µ is a probability measure, then the limit in (5.114) is also an L1 ˇ “ ˇ ‰ “ ‰ limit, and Pµ f “ Eµ f ˇ I , f P L1 pΩ, F, µq, where Eµ f ˇ I denotes the conditional expectation on the σ-field of invariant events: I “ tA P F : S1A “ 1A u. Before we prove this theorem we insert a corollary. 5.67. Corollary. Let the notation and hypotheses be as in Theorem 5.66. Suppose that the operator S is ergodic in the sense that S1 “ 1, and Sf “ f , If µ pΩq “ 8, then f P L1 pΩ, F, µq, implies f “ constant µ-almost everywhere. ş 1 Pµ f “ 0, f P L pΩ, F, µq. If µ pΩq “ 1, then Pµ f “ f dµ, f P L1 pΩ, F, µq.. Proof. Let f P L1 pΩ, F, µq. Then SPµ f “ Pµ f , and so by ergodicity Pµ f is a constant µ-almost everywhere. If µ pΩq “ 8, then this constant must be zero, because Pµ f belongs to L1 pΩ, F, µq. If µ pΩq “ 1, then, by the L1 -version of Theorem 5.66, we have ż ż n´1 ż 1 ÿ k (5.115) S f dµ “ f dµ, Pµ f dµ “ lim nÑ8 n k“0 and the inequality in (5.115) completes the proof of Corollary 5.67.. . Proof of Theorem 5.66. Deﬁne for f P L1 pΩ, F, µq, and ´8 ă a ă b ă f by 8 the subset Ca,b # + n´1 n´1 ÿ ÿ 1 1 f “ lim inf S k f ă a ă b ă lim sup Skf . (5.116) Ca,b nÑ8 n nÑ8 n k“0 k“0. 354 Download free eBooks at bookboon.com.

(119) 3. APart TASTE OF ERGODIC THEORY Advanced stochastic processes: II. 355 Some related results. ” ı f Then belongs to F, and by Theorem 5.65 it follows that µ Ca,b ă 8. By Proposition 5.64 we see that f Ca.b. S1C f “ 1C Sf “ 1C f . a,b. a,b. a,b. (5.117). As in equality (5.102) in Theorem 5.65 we write gr for the maximal function f , we see corresponding to g P L1 pΩ, F, µq. Then, with C “ Ca,b n´1 n´1 ( 1 ÿ k 1 ÿ␣ S tpa ´ f q 1C u “ sup a ´ S k´1 f 1C n n n n k“0 k“0 ˜ ¸ n´1 1 ÿ k “ a ´ inf S f 1C , (5.118) n n k“0. ppa Č ´ f q 1C q “ sup. f we see that and so from (5.118) and the deﬁnition of C “ Ca,b ) ! C Ă ppa Č ´ f q 1C q ą 0 .. (5.119). The inclusion in (5.119) together with Theorem 5.65 yields ż ż ż pa ´ f q 1C dµ “ pa ´ f q 1C dµ ě 0. (5.120) pa ´ f q 1C dµ “ Č q1C qą0u C tppa´f ş As a consequence (5.120) implies f dµ ď aµ pCq. A similar reasoning shows C ) ! that C Ă ppf Č ´ bq 1C q ą 0 , and therefore, like in (5.120), ż ż pf ´ bq 1C dµ “ pf ´ bq 1C dµ ě 0, ´bq1C qą0u tppfČ ş ş and hence bµ pCq ď C f dµ. Since (5.120) entails ´C f dµ ¯ ď aµ pCq, we obtain f bµ pCq ď aµ pCq. Since b ą a and µ pCq we get µ Ca,b “ 0. The subset C0 deﬁned by + # n´1 n´1 ÿ ÿ 1 1 C0 “ lim inf S k f ă lim sup Skf nÑ8 n nÑ8 n k“0 k“0 can be written in the form. C0 “. ď. f Ca,b ,. ´8ăaăbă8, a,bPQ. where the symbol Q denotes the set of rational numbers. So, by what is proved above the set C0 can be covered by a countable collection of subsets of the form f Ca,b , ´8 ă a ă b ă 8, all of which have µ-measure 0. Whence, µ pC0 q “ 0. So the pointwise limit in (5.114) exists µ-almost everywhere. (a) Next we prove assertion (a), and therefore Pµ f belongs to L1 pΩ, F, µq. By Fatou’s lemma we have ˇ ż ż ˇˇ ż n´1 n´1 ˇ ÿ 1 1 ÿ ˇˇ k ˇˇ ˇ k ˇ |Pµ f | dµ “ ˇ lim S f ˇ dµ ď lim inf S f dµ nÑ8 n ˇ ˇnÑ8 n k“0 k“0 355 Download free eBooks at bookboon.com.

(120) Advanced stochastic processes: Part II 356 5. SOME RELATED RESULTS. Some related results. ż n´1 ż n´1 ż 1 ÿ ˇˇ k ˇˇ 1 ÿ ď lim inf |f | dµ “ |f | dµ. S f dµ ď lim inf nÑ8 n nÑ8 n k“0 k“0 (5.121) The inequality in (5.121) shows property (a). The fact that Pµ2 “ Pµ follows from (b). First let f ě 0 belong to L1 pΩ, F, µq. Then n´1 n´1 1 ÿ k 1 ÿ k S f “ sup inf min 1 S f, Pµ f “ lim inf 1 nÑ n k“0 N N ěN N ďnďN n k“0. and hence. SPµ f “ lim inf nÑ. “ lim inf nÑ8. n1 n´1 1 ÿ k 1 ÿ k`1 S f “ sup inf min S f 1 1 n k“0 N N ěN N ďnďN n k“0 n´1 1 ÿ k`1 S f “ Pµ Sf “ Pµ f, n k“0. which implies property (b) for non-negative functions in L1 pΩ, F, µq. A general function can be written as a diﬀerence of non-negative functions in L1 pΩ, F, µq. This proves property (b).. Brain power. By 2020, wind could provide one-tenth of our planet’s electricity needs. Already today, SKF’s innovative knowhow is crucial to running a large proportion of the world’s wind turbines. Up to 25 % of the generating costs relate to maintenance. These can be reduced dramatically thanks to our systems for on-line condition monitoring and automatic lubrication. We help make it more economical to create cleaner, cheaper energy out of thin air. By sharing our experience, expertise, and creativity, industries can boost performance beyond expectations. Therefore we need the best employees who can meet this challenge!. The Power of Knowledge Engineering. Plug into The Power of Knowledge Engineering. Visit us at www.skf.com/knowledge. 356 Download free eBooks at bookboon.com. Click on the ad to read more.

(121) Advanced stochastic processes: Part II 4. PROJECTIVE LIMITS OF PROBABILITY DISTRIBUTIONS. Some related results 357. Next we assume that µ pΩq “ 1. First we will show that the pointwise limit in (5.114) is in fact also an L1 -limit. For this purpose we ﬁx f P L1 pΩ, F, µq and a real number M ą 0. Then we have ˇ ż ˇˇ n´1 ˇ ˇ1 ÿ k ˇ S f ´ Pµ f ˇ dµ ˇ ˇ n k“0 ˇ ˇ ˇ ˇ ¸ ¸ ˜ ż ˇ ż ˇˇ˜ n´1 n´1 ˇ ˇ ÿ ÿ ` ` ˘ ˘ 1 1 ˇ ˇ ˇ ˇ S k ´ Pµ f 1t|f |ăM u ˇ dµ ` ˇ S k ´ Pµ f 1t|f |ěM u ˇ dµ ď ˇ ˇ ˇ ˇ n k“0 ˇ n k“0 ˇ ˇ ¸ ˜ ż ˇ ż n´1 ` ˘ˇˇ ˇ 1 ÿ k S ´ Pµ f 1t|f |ăM u ˇ dµ ` 2 |f | dµ. (5.122) ď ˇ ˇ ˇ n k“0 t|f |ěM u. Since |g| ď M , g P L1 pΩ, F, µq implies |Sg| ď M and |Pµ g| ď M , the integrand in the ﬁrst term of the right-hand side of (5.122) is dominated by the constant L1 -function 2M . So from Lebesgue’s dominated convergence theorem, (5.114) and (5.122) it follows that ˇ ż ˇˇ n´1 ż ˇ ÿ ˇ ˇ1 k lim sup ˇ S f ´ Pµ f ˇ dµ ď 2 |f | dµ. (5.123) ˇ ˇ n k“0 nÑ8 t|f |ěM u Since M ą 0 is arbitrary and f P L1 pΩ, F, µq the inequality in (5.123) implies ˇ ż ˇˇ n´1 ˇ ÿ ˇ ˇ1 k lim sup ˇ S f ´ Pµ f ˇ dµ “ 0, ˇ ˇ n k“0 nÑ8. 1 which is the same as ˇ ‰ that1the limit in (5.114) also holds in L -sense. The “ saying equality Pµ f “ Eµ f ˇ I , f P L pΩ, F, µq, follows from the following two facts:. (a) the collection tA P F : S1A “ 1A u is a σ-ﬁeld, which is readily established. (b) Moreover, if f ě 0 is such that Sf “ f , and if α ą 0, then S1tf ąαu “ 1tSf ąαu “ 1tf ąαu .. This completes the proof of Theorem 5.66.. . 4. Projective limits of probability distributions This section is dedicated to a proof of Kolmogorov’s extension theorem. We will also present Carath´edory’s extension theorem. Let I be an arbitrary set of indices. Denote by H pIq the class of all ﬁnite subsets of I, by H1 pIq the collection of all countable subsets of I, and by H2 pIq “ 2I the class of all subsets of I. Consider a collection ofśmeasurable spaces pΩi , Ai q indexed by i P I. For J P H2 pIq we write ΩJ “ jPJ Ωj , and for J Ă K Ă I we denote K by pK J the canonical projection of ΩK onto ΩJ . If J “ tju Ă K we write pj I instead of pK tju , and if K “ I we write pJ instead of PJ . Hence pj denotes the (one-dimensional-)projection of ΩI on its j-th coordinate Ωj . Often these coordinate functions tpj : j P Iu serve as a canonical stochastic process. On. 357 Download free eBooks at bookboon.com.

(122) Advanced stochastic processes: Part II 358 5. SOME RELATED RESULTS. Some related results. each ΩJ , J P H␣2 pIq, we consider the σ-ﬁeld AJ “ bjPJ Aj , generated by the set ( J of projections pj : j P J , i.e. the smallest σ-ﬁeld containing the sets !` ˘ ) ␣␣ ( ( ´1 pJj pAq : j P J, A P AJ “ pJj P A : j P J, A P AJ .. The collection AJ is called the product σ-algebra on ΩJ . One easily sees that, if J P H pIq, then AJ is generated by the set of rectangles, i.e. by # + ź ź Aj “ Aj : Aj P Aj , j P J . jPJ. jPJ. If J Ă K Ă L Ă I, then we clearly have. L pLJ “ pK J ˝ pK .. (5.124). 2 It is easily seen that the projection pK J , where K, J P H pIq, J Ă K, from ΩK onto ΩJ is measurable for the σ-ﬁelds AK and AJ . The latter is also written as: pK J is AK -AJ -measurable. On ΩI we consider two classes of subsets: ␣ ( pAq : J P H pIq , A P A B “ tpJ P Au “ p´1 , and (5.125) J J ␣ ( 1 ´1 1 B “ tpJ P Au “ pJ pAq : J P H pIq , A P AJ . (5.126). The subsets belonging to B are called cylinders or cylinder sets. If Z P B, respectively Z P B1 , then there exists J P H pIq, respectively J 1 P H1 pIq, such that (5.127) Z “ A ˆ ΩIzJ . The inclusions B Ă B1 Ă AI are obvious.. 5.68. Definition. Let B be a subset of the powerset of ΩI . Then B is called a Boolean algebra, if it is closed under ﬁnite union, and under taking complements. 5.69. Lemma. The set B is a Boolean algebra, B1 is a σ-field, and σ tBu “ B1 “ AI .. (5.128). Proof. First we show that B is a Boolean algebra. Let Z “ ppJ q´1 pAq “ tpJ P Au, J P H pIq, A P AJ , be a cylinder. Then c Z c :“ ΩI zZ “ ΩI z ppJ q´1 pAq “ tpJ P ΩJ zAu “ p´1 J pA q ,. which shows that Z c belongs to B whenever Z P B. Furthermore, let Zi “ p´1 Ji pAi q, Ji P H pIq, Ai P AJi , i “ 1, . . . , n, be n cylinders. Then for J “ J1 Y ¨ ¨ ¨ Y Jn we have ´1 Z1 Y ¨ ¨ ¨ Y Zn “ p´1 J1 pA1 q Y ¨ ¨ ¨ Y pJn pAn q ` J ˘´1 ` J ˘´1 pA1 q Y ¨ ¨ ¨ Y p´1 pAn q “ p´1 pJ 1 pJ n J J ´` ˘ ¯ ` ˘ ´1 ´1 “ p´1 pJJ1 pA1 q Y ¨ ¨ ¨ pJJn pAn q . J. (5.129). Since the sets p´1 Ji pAi q belong to AJ for i “ 1, . . . , n the set Z1 Y ¨ ¨ ¨ Zn is a cylinder. In the same manner one proves that B1 is a σ-ﬁeld.. 358 Download free eBooks at bookboon.com.

(123) Advanced stochastic processes: Part II 4. PROJECTIVE LIMITS OF PROBABILITY DISTRIBUTIONS. Some related results 359. In order to get the equalities in (5.128) it remains to show that AI Ă σ tBu. Considering the deﬁnition of AI it is suﬃcient to prove that pi , i P I, is measurable for σ tBu and AI . However, this follows from (5.125). So the proof of Lemma 5.69 is complete now. 5.70. Remark. The fact that B1 “ AJ is important. It shows that each B P AI only depends on at most a countable number of indices, in the sense that B can be written as B “ A ˆ ΩIzJ where J is countable or ﬁnite and where A P AJ . The observation in this remark shows that the product σ-ﬁeld is relatively “poor” when the index set I is uncountable. The following two examples will clarify this. 5.71. Example. Take I uncountable, let each Ωi , i P I, be an arbitrary topological Hausdorﬀ space with at least two points, and let Ai be the Borel ﬁeld of Ωi . For every i P I we select ωi P Ωi . Since the singleton tpωi qiPI u is a closed subset of ΩI with respect to the product topology, it belongs to the Borel σ-ﬁeld of ΩI . But it does not belong to AI because it cannot be written as the set B in Remark 5.70.. 359 Download free eBooks at bookboon.com. Click on the ad to read more.

(124) Advanced stochastic processes: Part II 360 5. SOME RELATED RESULTS. Some related results. 5.72. Example. Take I “ r0, 8q and suppose that Ωi “ Ω, where Ω is a topological Hausdorﬀ space consisting of at least two points. Hence Ωr0,8q “ Ωr0,8q is the set of all mappings from r0, 8q to Ω. Let B be the subset of Ωr0,8q consisting of all right-continuous (or all continuous) mappings from r0, 8q to Ω. Assuming that B belongs to Ar0,8q “ Abr0,8q , where A is the Borel σ-ﬁeld of Ω will lead to a contradiction. Because, if B belongs to Ar0,8q , then by Remark 5.70 B is of the form B “ A ˆ Ωr0,8qzJ where J Ă r0, 8q is countable, and where A P AJ . We may suppose that J contains all rational numbers. Pick f P B and t P r0, 8q zJ. We deﬁne the function g : r0, 8q Ñ Ω as follows: gpsq “ f psq if s ‰ t, and gpsq ‰ f ptq if s “ t. Then g P B, but it is not right-continuous, which can be seen as follows. In J there exists a sequence ptn qn which decreases to t. Then lim g ptn q “ lim f ptn q “ f ptq ‰ gptq. nÑ8. nÑ8. It follows that B R Ar0,8q .. 5.73. Definition. Consider a family of measurable spaces pΩi , Ai q, i P I. Suppose that for every J P H pIq PJ is a probability measure on pΩJ , AJ q such that ”` ˘ ı ‰ “ K K ´1 PK pJ P A “ PK pJ pAq “ PJ rAs , (5.130). whenever J, K P H pIq, J Ă K, and A P AJ . Then the family tPJ : J P H pIqu, or the family tpΩJ , AJ , PJ q : J P H pIqu, is called a projective system of probability measures, or spaces. Such a system is also called a consistent system, or a cylindrical measure.. The following theorem says that a cylinder measure is a genuine measure provided that the spaces Ωi are topological Hausdorﬀ spaces which are Polish, endowed with their Borel σ-ﬁelds Bi . From Theorem 5.51 it follows that all probability measures µ on a Polish space S are inner and outer regular in the sense that µpBq “ sup tµpKq : K Ă B, K compactu “ inf tµpOq : O Ą B, O openu (5.131) whenever B belongs to the Borel σ-ﬁeld of S. The following theorem is a slight reformulation of Theorem 3.1. We also make the following observations. A second-countable locally-compact Hausdorﬀ space is Polish. See Theorem 1.16, and see the formula in (1.18) which gives the metric. As mentioned earlier this construction can be found in Garrett [57]. A countable disjoint union of Polish spaces pEj , dj q is Polish, with metric # 1, pfor x, y in distinct spaces in the unionq, dpx, yq “ (5.132) dn px, yq pfor x, y in the nth space in the unionq.. Here we assume that dj px, yq ď 1, x, y P Ej . From this result it follows that a σ-compact metrizable Hausdorﬀ space E “ Y8 j“1 Kj , Kj Ă Kj`1 , Kj compact, is a Souslin space, i.e. a continuous image of a Polish space. This is so because every subset Kj is compact metrizable, and therefore separable. Therefore the complements Kj`1 zKj , j P N, are Polish, and since E is the disjoint union of. 360 Download free eBooks at bookboon.com.

(125) Advanced stochastic processes: Part II 4. PROJECTIVE LIMITS OF PROBABILITY DISTRIBUTIONS. Some related results 361. such spaces E itself is Polish. It is known that probability measures on the class of Borel subsets of Souslin spaces are regular. For details the reader is referred to Bogachev [21]. Before we formulate and prove the Kolmogorov’s extension theorem we will discuss the Carath´eodory’s extension theorem. We need the notion of semi-ring, ring, and (Boolean) algebra of subsets of a given set Ω. 5.74. Definition (Deﬁnitions). Let Ω be a given set. A semi-ring is a subset S of PpΩq, the power set of Ω, which has the following properties: (i) H P S; (ii) For all A, B P S, the intersection A X B belongs to S (S is closed under pairwise intersections); (iii) For all A, B P S, there Ťn exist disjoint sets Ki P S, with i “ 1, 2, . . . , n, such that AzB “ i“1 Ki (relative complements can be written as ﬁnite disjoint unions). A ring R is a subset of the power set of Ω which has the following properties: (i) H P R; (ii) For all A, B P R, the union A Y B belongs to R (R is closed under pairwise unions); (iii) For all A, B P R, the relative complement AzB belongs to R (R is closed under relative complements). Thus any ring on Ω is also a semi-ring. A Boolean algebra B is deﬁned as a subset of the power set of Ω with the following properties: (i) H P B; (ii) For all A P B and B P B the union A Y B belongs to B; (iii) If A belongs to B, then its complement Ac “ ΩzA belongs to B. Sometimes, the following constraint is added in the measure theory context: Ω is the disjoint union of a countable family of sets in S. Without proof we mention some properties. Arbitrary (possibly uncountable) intersections of rings on Ω are still rings on Ω. If A is a non-empty subset of PpΩq, then we deﬁne the ring generated by A (noted R pAq) as the smallest ring containing A. It is straightforward to see that the ring generated by A is equivalent to the intersection of all rings containing A. For a semi-ring S, the set containing all ﬁnite disjoint union of sets of S is the ring generated by S: + # n ď A i , Ai P S . R pSq “ A : A “ i“1. This means that R pSq is simply the set containing all ﬁnite unions of sets in S.. 361 Download free eBooks at bookboon.com.

(126) Advanced stochastic processes: Part II 362 5. SOME RELATED RESULTS. Some related results. A content µ deﬁned on a semi-ring S can be extended on the ring generated by S. Such an extension is unique. The extended content is necessarily given by: n n ÿ ď µpAi q for A “ Ai , with the Ai P S’s mutually disjoint. µpAq “ i“1. i“1. In addition, it can be proved that µ is a pre-measure if and only if the extended content is also a pre-measure, and that any pre-measure on R pSq that extends the pre-measure on S is necessarily of this form.. Some motivation is at place here. In measure theory, one is usually not interested in semi-rings and rings themselves, but rather in σ-algebras (or σ-ﬁelds) generated by them. The idea is that it is possible to build a pre-measure on a semi-ring S (for example Stieltjes measures), which can then be extended to a pre-measure on R pSq, which can ﬁnally be extended to a genuine measure on a σ-algebra through Carath´eeodory’s extension theorem. As σ-algebras generated by semi-rings and rings are the same, the diﬀerence does not really matter (in the measure theory context at least). Actually, the Carath´eodory’s extension theorem can be slightly generalized by replacing ring with semi-ring.. Challenge the way we run. EXPERIENCE THE POWER OF FULL ENGAGEMENT… RUN FASTER. RUN LONGER.. RUN EASIER…. READ MORE & PRE-ORDER TODAY WWW.GAITEYE.COM. 1349906_A6_4+0.indd 1. 22-08-2014 12:56:57. 362 Download free eBooks at bookboon.com. Click on the ad to read more.

(127) Advanced stochastic processes: Part II 4. PROJECTIVE LIMITS OF PROBABILITY DISTRIBUTIONS. Some related results 363. 5.75. Definition. Let S be a semi-ring in P pΩq. A pre-measure on S is a map µ : S Ñ r0, 8s such that (i) µ pHq “ 0. Ť (ii) If pAn qn is a mutually disjoint sequence in S, and if A :“ n An belongs N ÿ ř to S, then µpAq “ n µ pAn q “ lim µ pAn q. N Ñ8. n“1. We also need the concept of outer or exterior measure. 5.76. Definition. An outer measure on P pΩq is a map λ : P pΩq Ñ r0, 8s with the following properties: (i) λ pHq “ 0. (ii) A Ă B implies λpAq ď λpBq. Ť ř (iii) If pAn qn is a sequence in P pΩq, then λ p n An q ď n λ pAn q.. By taking all but ﬁnitely many An to be the empty set one sees that an outer measure is sub-additive: λ pA Y Bq ď λpAq ` λpBq, A, B P P pΩq. Let λ be an outer measure on P pΩq. We deﬁne Σλ to be the set of all subsets A Ă Ω such that for any D Ă Ω we have λpDq “ λ pA X Dq ` λ pAc X Dq .. (5.133). λpDq ě λ pA X Dq ` λ pAc X Dq .. (5.134). Since an outer measure λ is sub-additive we may replace the equality in (5.133) be an inequality of the form In other words, Σλ consists of all subsets A Ă Ω that split Ω in two in a good way. Clearly, Ω P Σλ and by the very form of the deﬁnition of Σλ , we have a subset A belongs to Σλ if and only if its complement Ac belongs to Σλ . We now present the following proposition, whose proof is a bit tedious. For details the reader is referred to, e.g., [5] or [10]. The reader may also want to consult the Probability Tutorials by Noel Vaillant: The sets in Σλ are called Carath´eodory measurable relative to the outer measure λ. 5.77. Proposition. Let λ be an outer measure on Ω, and let Σλ be as defined above. Then Σλ is a σ-algebra on Ω. şb The Lebesgue-Stieltjes integral a f pxq dgpxq is deﬁned when f : ra, bs Ñ R is Borel-measurable and bounded and g : ra, bs Ñ R is of bounded variation in ra, bs and right-continuous, or when f is Borel-measurable and non-negative and g is non-decreasing, and right-continuous. Deﬁne wpps, tsq :“ gptq ´ gpsq and wptauq :“ 0 (Alternatively, the construction works for g left-continuous, wprs, tqq :“ gptq ´ gpsq and wptbuq :“ 0). By Carath´eodory’s extension theorem (Theorem 5.79), there is a unique Borel measure µg on ra, bs which agrees with w on every interval I Ă ra, bs. The measure µg arises from the outer measure # + ÿ ď µg pEq “ inf µg pIi q : E Ă Ii , i. i. 363 Download free eBooks at bookboon.com.

(128) Advanced stochastic processes: Part II 364 5. SOME RELATED RESULTS. Some related results. where the inﬁmum is taken over all coverings of E by countably many semi-open or Stieltintervals Ii . This measure is sometimes called the Lebesgue-Stieltjes, şb jes measure associated with g. The Lebesgue-Stieltjes integral a f pxq dgpxq is deﬁned as the Lebesgue integral of f with respect to the measure µg in the usual şb şb way. If g is non-increasing, then deﬁne a f pxq dgpxq :“ ´ a f pxq dp´gqpxq. If the function g : ra, bs Ñ R is right-continuous, and of bounded variation on ra, bs, then g may be written in the form g “ g1 ´ g2 , where the functions g1 and g2 are monotone non-decreasing and right-continuous. So that µg ps, ts “ gptq ´ gpsq, a ď s ă t ď b, extends to a real-valued measure on the Borel ﬁeld of ra, bs. Of course, if g were right-continuous, complex-valued and of bounded variation, then g can be split as follows g “ Re g ` iIm g “ g1 ´ g2 ` i pg3 ´ g4 q where the functions gj , 1 ď j ď 4, are right-continuous, and non-decreasing. To the function g we can associate a complex-valued measure µg such that µg ps, ts “ gptq ´ gpsq, a ď s ă t ď b. For more details on Riemann-Stieltjes integrals the reader is referred to [130]. The book by Tao [137] contains a discussion on Stieltjes measures. The deﬁnition of semi-ring may seem a bit convoluted, but the following simple example shows why it is useful. 5.78. Example. Think about the subset of P pRq deﬁned by the set of all halfopen intervals pa, bs for a and b reals. This is a semi-ring, but not a ring. Stieltjes measures are deﬁned on intervals; the countable additivity on the semi-ring is not too diﬃcult to prove because we only consider countable unions of intervals which are intervals themselves. Proving it for arbitrary countably union of intervals is proved using Carath´eodory’s extension theorem. Now we are ready to formulate the Carath´eodory’s extension theorem. 5.79. Theorem (Carath´eodory’s extension theorem). Let R be a ring on Ω and µ : R Ñ r0, 8s be a pre-measure on a R. Then there exists ˇ a measure µ1 : σ pRq Ñ r0, 8s such that µ1 is an extension of µ. (That is, µ1 ˇR “ µ). Here σ pRq is the σ-algebra generated by R. If µ is σ-finite then the extension µ1 is unique (and also σ-finite).. If R is a Boolean algebra, then Theorem 5.79 is also called the Hahn-Kolmogorov extension theorem. A complete proof can also be found in [21] Theorem 1.5.6. We will present just an outline. Another interesting book is Tao [137]; in particular see Theorems 1.7.3 (Carath´eodory’s extension theorem) and 1.7.8 together with Exercise 1.7.7 (Hahn-Kolmogorov’s extension theorem). An (older) paper, which treats Carath´eodory’s extension theorem thoroughly, is Maharam [90]. Proof. The proof is based on the σ-ﬁeld corresponding to the outer (or exterior) measure associated to pre-measure µ. This exterior measure µ˚ is deﬁned by # + 8 8 ÿ ď µ pAk q : Ak P R, A Ă Ak , A Ă Ω. (5.135) µ˚ pAq “ inf k“1. k“1. 364 Download free eBooks at bookboon.com.

(129) Advanced stochastic processes: Part II 4. PROJECTIVE LIMITS OF PROBABILITY DISTRIBUTIONS. Some related results 365. (If A can not be covered by a countable union of sets in R, then we put µ˚ pAq “ 8.) Then it is not too diﬃcult to prove that µ˚ is an outer measure. Like in Proposition 5.77 let Σµ˚ be the σ-ﬁeld consisting of those subsets A of Ω for which µ˚ pDq ě µ˚ pA X Dq ` µ˚ pAc X Dq for all D Ă Ω. Then it follows that the σ-ﬁeld Σµ˚ contains the ring R. Put µ1 pBq “ µ˚ pBq, B P Σµ˚ . Then µ1 is a measure on Σµ˚ which extends µ, and which unique provided that µ is σ-ﬁnite. For details see [21] Theorem 1.5.6. This concludes an outline of the proof of Theorem 5.79. 5.80. Example. Let E be a σ-compact topological Hausdorﬀ space, and assume that each compact subset Kj is metrizable, and hence separable. Deﬁne the sequence of open subsets pOj qj of E as follows: O0 “ H, O1 “ EzK1 , Oj`1 “ pEzK1 qX¨ ¨ ¨XpEzKj q, j ě 1. Then, for an appropriate metric px, yq ÞÑ dj px, yq, x, y P Kj X Oj , 0 ď dj px, yq ď 1, the spaces Kj X Oj is complete metrizable and separable, and so a Polish space. Moreover, by construction the spaces Kj X Oj , j “ 0, 1, . . ., are mutually disjoint, and so the E can be supplied with the metric dpx, yq deﬁned by dpx, yq “ 1, if x, y belong to diﬀerent spaces Kj X Oj , and dpx, yq “ dj px, yq, if x and y belong to Kj X Oj , j “ 0, 1, . . .. Then this metric turns E written as a disjoint union of Kj X Oj into a Polish space. Its topology is stronger than the original one, and hence E itself is continuous image of a Polish space (via the identity map). It follows that E is a Souslin space.. This e-book is made with. SETASIGN. SetaPDF. PDF components for PHP developers. www.setasign.com 365 Download free eBooks at bookboon.com. Click on the ad to read more.

(130) Advanced stochastic processes: Part II 366 5. SOME RELATED RESULTS. Some related results. Example 5.80 should be compared with the notion of disjoint unions of Polish spaces are again Polish: see (5.132). 5.81. Theorem (Kolmogorov’s extension theorem). Let tpΩJ , BJ , PJ q : J P H pIqu. be a projective system of probability spaces. Suppose that for every i P I, Ωi is a Polish space (or Souslin space) endowed with its Borel σ-field Bi . Then there exists a unique probability measure PI on pΩI , BI q such that ‰ “ PI rpJ P As “ PI p´1 (5.136) J pAq “ PJ rAs , A P BJ ,. for every J P H pIq.. Proof. If Z “ tpJ P Au “ p´1 J pAq, J P H pIq, A P BJ , is a cylinder in ΩI , then we deﬁne PI rZs by PI rZs “ PJ rAs . (5.137) This deﬁnition is unambiguous. Indeed, let ´1 Z “ tpJ P Au “ p´1 J pAq “ pK pBq “ tpK P Bu , with A P BJ and B P BK ,. with J, K P H pIq. We have to show that PJ rAs “ PK rBs. Indeed, with L “ J Y K, we get ␣ ( ` L ˘´1 pAq “ pLJ ˝ pL P A Z “ p´1 pJ L ( ` ˘´1 ( ␣ ` ˘´1 ␣ pAq “ pLK pBq “ pLK P B “ pLJ P A “ pLJ ␣ ( ` L ˘´1 “ pLK ˝ pL P B “ p´1 pBq . (5.138) pK L From (5.130) together with (5.138) we infer ”` ˘ ”` ˘ ı ı ´1 ´1 pAq “ PL pLK pBq “ PK rBs . PJ rAs “ PL pLJ. The equality in (5.139) shows that PI is well deﬁned. We also have ‰ “ PI rΩI s “ PI p´1 i pΩi q “ Ptiu rΩi s “ 1.. (5.139). Next we show that PI is ﬁnitely additive on B, the collection of cylinders. Let ´1 1 Z “ p´1 J pAq, with J P H pIq and A P BJ , and Z “ pK pBq, with K P H pIq and B P BK be two disjoint cylinders. Put L “ J Y K. Then we have ` L ˘´1 ` L ˘´1 pAq X p´1 pBq H “ Z X Z 1 “ p´1 pJ pK L L ´` ˘ ¯ ` ˘ ´1 ´1 “ p´1 pLJ pAq X pLK pBq . (5.140) L ` ˘´1 ` ˘´1 pBq “ H. Consequently, we obtain pAq X pLK From (5.140) we infer pLJ ” ı ` L ˘´1 ` L ˘´1 ´1 pAq Y p pBq PI rZ Y Z 1 s “ PI p´1 p p J K L L ” ´` ˘ ¯ı ` ˘ L ´1 L ´1 p pAq Y p pBq “ PI p´1 J K L ”` ˘ ı ` ˘´1 ´1 pAq Y pLK pBq “ PL pLJ ”` ˘ ”` ˘ ı ı ´1 ´1 pAq ` PL pLK pBq “ PL pLJ 366 Download free eBooks at bookboon.com.

(131) Advanced stochastic processes: Part II 4. PROJECTIVE LIMITS OF PROBABILITY DISTRIBUTIONS. Some related results 367. (apply the equality in (5.130)) “ PJ rAs ` PK rBs “ PI rAs ` PI rBs .. (5.141). The equality in (5.141) proves the ﬁnite additivity of the mapping PI on the collection of cylinder sets B. Finally we prove that the mapping PI is σ-additive on B. For that purpose we of cylinder sets such that PI rZn s ě a ą 0 consider a decreasing sequence pZn qn Ş for all n P N. Ş We will show that n Zn ‰ H. By contraposition it then follows that n Zn “ H implies limnÑ8 PI rZn s “ 0. For each n we have Zn “ p´1 An P BJn . Of course we may suppose that J pAn q with Jn P H pIq and Ť J1 Ă J2 Ă ¨ ¨ ¨ Ă Jn Ă ¨ ¨ ¨ . Put J “ n Jn . Then ` ˘´1 ` J ˘´1 Zn “ p´1 pAn q “ pJJn pAn q ˆ ΩIzJ . pJ n J Since. č n. Zn “. ˜. č`. pJJn. n. ˘´1. ¸. pAn q. ˆ ΩIzJ. (5.142). Ş ` ˘´1 Ş pAn q ‰ H. This means that we see that n Zn ‰ H if and only if n pJJn our problem is reduced to the problem with I “ J, i.e. to a countable problem. For every m P N there exists a compact subset Ljm of Ωjm with a Pjm rΩjm zLjm s ď . 4 ˆ 2m ś Then L :“ m Ljm is a compact subset of ΩJ . Furthermore, for every n P N we have ¸c ﬀ « «˜ ﬀ ď ` ˘´1 ` ˘ ź ÿ a Pj rΩj zLj s ă . (5.143) Lj PJn “ PJn pJj n Lcj ď 4 jPJ jPJ jPJ n. n. n. On the other hand for every n P N we choose a compact subset Kn of An (in BJn ) such that a PJn rAn zKn s ď . (5.144) 4 ˆ 2n For every n P N the set Yn deﬁned by ¯´1 ´ ` ˘´1 Yn “ pJJn1 pK1 q X ¨ ¨ ¨ X pJJnn´1 pKn´1 q X Kn. is a closed subset ΩJn , and so Zn1 :“ p´1 Jn pYn q is a closed cylinder in ΩJ , and 1 Zn Ă Zn . In addition, we have ´1 ´1 Zn1 “ p´1 J1 pK1 q X ¨ ¨ ¨ X pJn´1 pKn´1 q X pJn pKn q. the sequence pZn1 qn is decreasing. We also have ˜ ¸ﬀ « č p´1 PI rZn zZn1 s “ PI Zn z Jk pKk q 1ďkďn. ď. n ÿ. k“1. EI. “. Zn zp´1 Jk. ‰. pKk q ď. n ÿ. k“1. ‰ “ EI Zk zp´1 pK q k Jk. 367 Download free eBooks at bookboon.com.

(132) Advanced stochastic processes: Part II 368 5. SOME RELATED RESULTS. “ “. n ÿ. k“1 n ÿ. Some related results. n ‰ ÿ ‰ “ “ ´1 EI p´1 pA q zp pK q “ EI p´1 k k Jk Jk Jk pAk zKk q k“1. a EJk rAk zKk s ă . 4 k“1. (5.145). Since, by assumption, PI rZn s ě a, (5.145) implies PJn rYn s “ PI rZn1 s “ PI rZn s ´ PI rZn zZn1 s ą a ´. 3a a “ . 4 4. (5.146). Since, by (5.146) and (5.143) we have « ﬀ «˜ ¸c ﬀ čź ź a 3a a ´ “ , (5.147) Lj Lj ě 1 ´ PJn rYnc s ´ PJn ě PJn Yn 4 4 2 jPJ jPJ n. n. Şś. Lj ‰ H. Moreover, observe that ˜ ¸ ˜ ¸ č ź čź ź ` ˘ L j m “ Yn Lj ˆ Lj , (5.148) Zn1 X L “ Yn ˆ ΩJzJn. it follows that Yn. jPJn. m. jPJn. jPJzJn. and consequently, Zn1 X L ‰ H. Hence, the decreasing sequence pZn1 X Lqn consists of non-empty compact subsets of ΩJ . By compactness we get that Ş 1 n Zn X L ‰ H. So we infer č č č 1 p´1 pA q Ą Z Ą pZn1 X Lq ‰ H. (5.149) n n Jn n. n. n. As a consequence of the previous arguments, we see that PI is a σ-additive on the Boolean algebra B which consists of cylinders in ΩI . This measure PI satisﬁes (5.136). By the classical Carath´eodory theorem the mapping PI extends in a unique fashion as a probability measure on the σ-ﬁeld σ tBu “ BI . Then, technically speaking, the mapping PI , deﬁned on the Boolean algebra B, is a pre-measure. This corresponding exterior measure P˚I is deﬁned by # + 8 8 ÿ ď µ pZk q : Zk P B, A Ă Zk . (5.150) P˚I pAq “ inf k“1. k“1. Then it is not so diﬃcult to prove that the set function deﬁned by (5.150) is an outer measure indeed. Deﬁne the associated σ-ﬁeld D by D “ tA Ă ΩI : P˚I pDq ě P˚I pA X Dq ` P˚I pAc X Dq : for all D Ă ΩI u . (5.151) The fact that D is a σ-ﬁeld indeed follows from Proposition 5.77: see Theorem 5.79 as well. It is fairly easy to see that D contains the Boolean algebra B which consists of the cylinder sets in ΩI . This completes the proof of Theorem 5.81.. 368 Download free eBooks at bookboon.com. .

(133) Advanced stochastic processes: 5. Part II UNIFORM INTEGRABILITY. Some related results 369. 5. Uniform integrability The next Theorem is often used as a replacement for the dominated convergence theorem of Lebesgue. 5.82. Theorem (Theorem of Scheﬀ´e). Let pΩ, F, µq be an arbitrary measure space and let pfn : n P Nq be a sequence of non-negative functions in L1 pΩ, F, µq. In addition, let the function f belong to L1 pΩ, F, µq. Suppose that f pxq “ limnÑ8 fn pxq for µ-almost all x P Ω. The following assertions are equivalent: ş (i) limnÑ8 |fn ´ f | dµ “ 0; (ii) The sequence pfnş: n P Nq is uniformly integrable; ş (iii) limnÑ8 fn dµ “ f dµ.. Instead of uniformly integrable the term equi-integrable is often used. A family if for every ϵ ą 0 there exists pfα : α P Aq in L1 pΩ, F, µq is uniformly integrable, ş 1 a function g ě 0 in L pΩ, F, µq such that tfα ěgu |fα | dµ ď ϵ for all α P A.. 5.83. Proposition. If µ is a probability measure, then a family pfα : α P Aq in L1 pΩ, F, µq is uniformly integrable, if and only if for every ϵ ą 0 there exists a ş constant Mε ě 0 such that t|fα |ěMε u |fα | dµ ď ϵ for all α P A.. www.sylvania.com. We do not reinvent the wheel we reinvent light. Fascinating lighting offers an infinite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges. An environment in which your expertise is in high demand. Enjoy the supportive working atmosphere within our global group and benefit from international career paths. Implement sustainable ideas in close cooperation with other specialists and contribute to influencing our future. Come and join us in reinventing light every day.. Light is OSRAM. 369 Download free eBooks at bookboon.com. Click on the ad to read more.

(134) Advanced stochastic processes: Part II 370. Some related results. 5. SOME RELATED RESULTS. Proof. The suﬃciency is clear: choose for gε a constant function Mε . Next we show that, if the family pfα : α P Aq is uniformly integrable, then necessarily for every ε ą 0 there exists a constant Mε such that ż |fα | dµ ď ε, α P A. (5.152) t|fα |ěMε u. Fix ε ą 0. By hypothesis we know that there exists a function gε P L1 pΩ, F, µq, gε ą 0, such that ż ε (5.153) |fα | dµ ď , α P A. 2 t|fα |ěgε u Then we choose Mε so large that ż ε (5.154) gε dµ ď . 2 tgε ěMε u Then by (5.153) and (5.154) we have ż ż ż |fα | dµ “ |fα | dµ ` |fα | dµ t|fα |ěMε u tMε ď|fα |ămaxpMε ,gε qu t|fα |ěmaxpMε ,gε qu ż ż ε ε ď gε dµ ` |fα | dµ ď ` “ ε. (5.155) 2 2 tgε ąMε u t|fα |ěgε u . The inequality in (5.155 completes the proof of Proposition 5.83.. Proof of Theorem 5.82. (i) ñ (ii). Put g “ supnPN fn . The following inequalities hold for m P N: ż ż ż fn dµ ď |fn ´ f | dµ ` f dµ tfn ěmf u tfn ěmf u tfn ěmf u ż ż ď |fn ´ f | dµ ` f dµ tfn ěmf u ż ż ď |fn ´ f | dµ ` f dµ. (5.156) tgěmf u. ş Let ϵ ą 0, but arbitrary. By (i) there exists N pϵq P N such that |fn ´ f | dµ ď ϵ{2 for n ě N pϵq ` 1. The inequalities below then follow for m ě M pϵq: ż ż f dµ ď ϵ{2, and fn dµ ď ϵ, 1 ď n ď N pϵq. (5.157) tgěmf u. tfn ěmf u. ş. From (5.156) and (5.157) we see tfn ěM pϵqf u fn dµ ď ϵ. But this means that the sequence pfn : n P Nq is uniformly integrable. (ii) ñ (iii). Let ϵ ą 0 be arbitrary and choose a function gϵ P L1 pΩ, F, µq such that ż ż fn dµ ` f dµ ď ϵ. (5.158) tfn ěgϵ u. From (5.158) we obtain ˇż ˇ ż ż ˇ ˇ ˇ fn dµ ´ f dµˇ ď ˇ ˇ. tfn ďgϵ u. tfn ěgϵ u. |fn ´ f | dµ `. ż. tfn ěgϵ u. fn dµ `. 370 Download free eBooks at bookboon.com. ż. tfn ěgϵ u. f dµ.

(135) Advanced stochastic processes: Part II 5. UNIFORM INTEGRABILITY. ď. ż. tfn ďgϵ u. Some related results 371. |fn ´ f | dµ ` ϵ.. (5.159). By the theorem of dominated convergence, it follows from (5.159) that ˇ ˇż ż ˇ ˇ ˇ lim sup ˇ fn dµ ´ f dµˇˇ ď ϵ. nÑ8. Since ϵ is arbitrary assertion (iii) follows. The same argumentation shows the implication (ii) ñ (i). (iii) ñ (i). The equality. |fn ´ f | “ fn ´ f ` 2 pf ´ min pf, fn qq. is obvious. From (iii) together with the theorem of dominated convergence it then follows that ż lim |fn ´ f | dµ nÑ8 ż ż “ lim pfn ´ f q dµ ` 2 lim pf ´ min pf, fn qq dµ “ 0. nÑ8. nÑ8. The proof of Theorem 5.82 is now complete.. . 5.84. Corollary. Let pµm : m P Nq be a sequence of probability measures on the Borel σ-field of Rν . Let every measure µm have a probability density gm relative to the Lebesgue measure λ. Furthermore, let g ě 0 be a probability density. Suppose that for λ-almost all x P Rν the equality limmÑ8 gm pxq “ gpxq is true. Let the measure µ have density g. Then the sequence pµm : m P Nq converges weakly to µ. Proof. From the theorem of Scheﬀ´e (Theorem 5.82) we see ż lim |gm pxq ´ gpxq| dx “ 0. mÑ8. Let f be a bounded continuous function. Then ˇż ˇ ˇż ˇ ż ż ˇ ˇ ˇ ˇ ˇ f dµm ´ f dµˇ “ ˇ pf pxqgm pxq ´ f pxqgpxqq dxˇ ď }f } |gm pxq ´ gpxq| dx. 8 ˇ ˇ ˇ ˇ (5.160) The assertion in Corollary 5.84 follows from (5.160). 5.85. Theorem. Let pXm : m P Nq be a sequence of stochastic variables, which are defined a probability space pΩ, F, Pq. (a) If the sequence pXm : m P Nq converges in probability to a stochastic variable X, then the sequence of probability measures pPXm : m P Nq converges weakly to the distribution PX ; (b) If the sequence pPXm : m P Nq converges vaguely to the Dirac-measure δa , then the sequence pXm : m P Nq converges in probability to a stochastic variable X, which is P-almost surely equal to the constant a.. 371 Download free eBooks at bookboon.com.

(136) Advanced stochastic processes: Part II 372. Some related results. 5. SOME RELATED RESULTS. Proof. (a) Suppose that the sequence pXm : m P Nq convergesş in probaν şbility to X. We pick f P C00 pR q and we şwill prove thatş limmÑ8 f dPXm “ f dPX . The latter is equivalent to limmÑ8 f pXm q dP “ f pXq dP. The function f is uniformly continuous. So, for ϵ ą 0 given, there exists δ ą 0 such that (5.161) |x2 ´ x1 | ď δ impliceert |f px2 q ´ f px1 q| ď ϵ. Put Am “ t|X ´ Xm | ě δu. For ω R Am the inequality |f pXm pωqq ´ f pXpωqq| ď ϵ. holds. it follows ˇ żthat ˇż From this ż ż ˇ ˇ ˇ f dPXm ´ f dPX ˇ ď |f pXq ´ F pXm q| dP ` ˇ ˇ c Am. Am. |f pXq ´ f pXm q| dP. ď ϵP pAcm q ` 2 }f }8 P t|Xm ´ X| ě δu. ď ϵ ` 2 }f }8 P t|Xm ´ X| ě δu .. 360° thinking. (5.162). The assertion in (a) follows from (5.162) together with assertion (3) in Theorem 5.43.. .. 360° thinking. .. 360° thinking. .. Discover the truth at www.deloitte.ca/careers. © Deloitte & Touche LLP and affiliated entities.. Discover the truth at www.deloitte.ca/careers. Deloitte & Touche LLP and affiliated entities.. © Deloitte & Touche LLP and affiliated entities.. Discover the truth 372 at www.deloitte.ca/careers Click on the ad to read more Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities.. Dis.

(137) Advanced stochastic processes: Part II. Some related results. 6. STOCHASTIC PROCESSES. 373. (b) Suppose that the sequence pPXm : m P Nq vaguely converges to the Diracmeasure δa . Let Ipϵq be the interval Ipϵq “ ra ´ ϵ, a ` ϵs and choose functions f and g P C00 pRν q such that f ď 1I ď g and such that f paq “ gpaq “ 1. Then the equalities follow: ż f paq “ lim inf f dPXm ď lim inf PXm pIq ď lim sup PXm pIq m m m ż (5.163) ď lim sup gdPXm “ gpaq. m. From (5.163) it follows that. lim P p|Xm ´ a| ď ϵq “ 1, m. which amounts to the same as lim P p|Xm ´ a| ą ϵq “ 0. m. This proves assertion (b). So the proof of Theorem 5.85 is now complete.. . 6. Stochastic processes We begin with some deﬁnitions. 5.86. Definition. Let pΩ, F, Pq be a probability space, and let pE, Eq be a locally compact Hausdorﬀ space, that satisﬁes the second countability axiom, with Borel σ-ﬁeld E. Often E will be chosen as R or as Rν . A stochastic process X with values in the state space E is a mapping X : r0, 8q ˆ Ω Ñ E. For every ω P Ω the mapping t ÞÑ Xpt, ωq deﬁnes a path of the process. A path is sometimes also called a realization. If we ﬁx n P N, then the mappings E b ¨ ¨ ¨ b E Ñ r0, 1s, where pt1 , . . . , tn q varies over r0, 8qn , and which Pt1 ,...,tn : looooomooooon nˆ. are deﬁned by. Pt1 ,...,tn pBq “ P tpXpt1 q, . . . , Xptn qq P Bq ,. BPE b ¨ ¨ ¨ b E, looooomooooon. (5.164). nˆ. are called the n-dimensional distributions of the process X. Here Xptq is the mapping Xptqpωq “ Xpt, ωq, ω P Ω. Sometimes we write Xt instead of Xptq. If n “ 1, then the distributions in (5.164) are also called the marginal distributions, or marginals. However, notice that a process is much more than the corresponding collection of ﬁnitedimensional distributions. In particular the paths or realizations of a process are very important. For example, the continuity properties of the paths are relevant. Often we will suppose that the paths are continuous, or that they are continuous from the right, and possess limits from the left {c`adl`ag paths}, or cadlag paths. So that the process X is cadlag provided that for all t ě 0 the equality limsÓt Xpsq “ Xptq holds P-almost surely (this is continuity from the right, or continue `a droite in French) and if the limit limsÒt Xpsq exists in E (this means that the left limits exist in E, limit´e `a gauche in French).. 373 Download free eBooks at bookboon.com.

(138) Advanced stochastic processes: Part II 374 5. SOME RELATED RESULTS. Some related results. 5.87. Definition. A family sub-σ-ﬁelds pFt : t ě 0q of F is called a filtration (or, sometimes, also called history), if t ă s implies Ft Ă Fs . Thus the probability P isŤdeﬁned on all σ-ﬁelds Ft . With F8 , or also F8´ the σ-ﬁeld Ş generated by tě0 Ft is meant. If for every t ě 0 the equality Ft “ sąt Fs holds, then the ﬁltration pFt : t ě 0q is called continuous from the Ş right, or rightcontinuous. Let pFt : t ě 0q be a ﬁltration, and put Ft` “ sąt Fs . Then the family pFt` : t ě 0q is a right-continuous ﬁltration. This ﬁltration is called the right closure of the ﬁltration pFt : t ě 0q. A subset A of Ω is called a P-null set if there exists a subset A0 P F with the following properties: A Ď A0 and PrA0 s “ 0. Usually this is expressed by saying that A is a null set instead of A is a P-null set. Often it is assumed that F0 contains all null sets, and that the ﬁltration pFt : t ě 0q is right-continuous. Sometimes it i said that F0 has the usual properties. The process X is called adapted to the ﬁltration pFt : t ě 0q if for every t ě 0 the state variable Xptq is measurable with respect to σ-ﬁelds Ft and E. Let Ht “ σ pXpuq : 0 ď u ď tq be the σ-ﬁeld generated by the state variables Xpuq, 0 ď u ď t. The ﬁltration pHt : t ě 0q is called the internal history of the process X. If t ą 0 is given, then Ht is called the (information from the) past, σ pXptqq is called the (information from the) present, and σ pXpuq : u ě tq the (information from the) future. The process X is adapted if and only if Ht Ď Ft for every t ě 0. 5.88. Definition. Let X and Y be two processes. The processes X and Y are said to be non-P-distinguishable or P-indistinguishable provided there exists a P-null subset N with the property that for every ω R N and for every t ě 0 the equality Xpt, ωq “ Y pt, ωq holds. The process X is called a modification of the process Y (or also Y is a modiﬁcation of X) if for every t ě 0 there exists a P-null set Nt with the property that Xpt, ωq “ Y pt, ωq for ω R Nt . Thus the null set is t-dependent. If the processes X and Y are not distinguishable, then X is a modiﬁcation of Y . In general, the converse statement is not true. 5.89. Theorem. Suppose that the process X as well as the process Y possesses right-continuous paths. If X is a modification of Y , then X and Y are not distinguishable (also called stochastically equivalent). Proof. Let X be a modiﬁcation of the process Y . For every t ě 0 there then exists a null set Nt such that Xptq “ Y ptq on the complement of Nt . Put Ť N “ tPQ Nt . Then PpN q “ 0 and for every t P Q the equality Xptq “ Y ptq holds on the complement of N . By right-continuity of the paths it then follows that Xptq “ lim Xpsq “ lim Y psq “ Y ptq sÓ0,sPQ. sÓt,sPQ. on the complement of N and completes the proof of Theorem 5.89.. . 5.90. Definition. Let pFt : t ě 0q be a ﬁltration and let T : Ω Ñ r0, 8s be a “stochastic time”. The function T is called a stopping time for the ﬁltration pFt : t ě 0q if for everyŤﬁxed time t the event tT ď tu belongs to Ft . Since the event tT ă 8u “ nPN tT ď nu belongs to F8 , the complementary event tT “ 8u is also an element of F8 .. 374 Download free eBooks at bookboon.com.

(139) Advanced stochastic processes: Part II 6. STOCHASTIC PROCESSES. Some related results 375. 5.91. Theorem. Let pFt : t ě 0q be a filtration. Let pFt` : t ě 0q be the so-called right closure of the filtration pFt : t ě 0q. Then a stochastic time T : Ω Ñ r0, 8s is a stopping time for the filtration pFt` : t ě 0q if and only if, for every t ą 0, the event tT ă tu belongs to Ft . Proof. “Suﬃciency” Suppose that for"every t ě 0*the event tT ă tu beč 1 longs to Ft . Then the event tT ď tu “ T ă1` belongs to the σ-ﬁeld n nPN Ş nPN Ft`n´1 “ Ft .. “Necessity” Assume that for every t ě 0 the * event tT ď tu belongs to Ft` . ď" Ť 1 T ď1´ belongs to nPN Ft´n´1 ` Ă Ft . Then the event tT ă tu “ n nPN This completes the proof of Theorem 5.91. . 5.92. Corollary. Let pFt : t ě 0q be a right-continuous filtration. Then the stochastic time T is a pFt : t ě 0q-stopping time if and only if for every t ě 0 the event tT ă tu belongs to Ft and this is the case for every t ą 0 if and only if for every t ą 0 the event tT ď tu belongs to Ft .. We will turn your CV into an opportunity of a lifetime. Do you like cars? Would you like to be a part of a successful brand? We will appreciate and reward both your enthusiasm and talent. Send us your CV. You will be surprised where it can take you.. 375 Download free eBooks at bookboon.com. Send us your CV on www.employerforlife.com. Click on the ad to read more.

(140) Advanced stochastic processes: Part II 376 5. SOME RELATED RESULTS. Some related results. 5.93. Theorem. Let pFt : t ě 0q be a right-continuous filtration, let X be an adapted cadlag process, let G be an open subset and let F be a closed subset of E. InŞaddition, let pGn : n P Nq be a sequence of open subsets of E such that : n P Nq be an F “ n Gn and such that Gn Ą Gn`1 , n P N. Finally, let pFn Ť increasing sequence of closed subsets with the property that G “ n Fn . Define the times S, Sn , T and Tn by means of the equalities: S “ inf ts ě 0 : Xpsq P F or Xps´q P F u ;. Sn “ inf ts ě 0 : Xpsq P Fn or Xps´q P Fn u ; Tn “ inf ts ě 0 : Xpsq P Gn u. and T “ inf ts ě 0 : Xpsq P Gu .. (5.165). Then these times are stopping times and the following assertions hold: Sn Ó T and Tn Ò S. Proof. Let t ą 0. Since the paths are continuous from the right se see ď ď tXprq P Gu “ tXprq P Gu P Ft . tT ă tu “ 0ărăt. 0ărăt,rQ. This proves that T is a stopping time. Since ˜ tS ď tu “ tXptq P F or Xpt´q P F u Y. č ď. nPN răt,rPQ. tXprq P Gn u. ¸. P Ft. it follows that S is a stopping time as well. Since Gn Ą Gn`1 it follows that Tn`1 ě Tn . Put S0 “ sup Tn . The ultimate equalities in 8 č 8 ď ď tS0 ă tu “ tXpsq P Gn u m“1 n“1 0ďsďt´m´1. “ “. 8 ď. ď. m“1 0ďsďt´m´1. ď. 0ďsăt. tXpsq P F of Xps´q P F u. tXpsq P F of Xps´q P F u “ tS ă tu. prove the equalities tS0 ă tu “ tS ă tu for all t ą 0 and hence, S “ S0 . The fact that Sn Ó T is left to the reader as an exercise. This completes the proof of Theorem 5.93. 5.94. Theorem. Let S and T be stopping times for the filtration pFt : t ě 0q. Then minpS, T q, maxpS, T q and S ` T are also stopping times for this filtration. If pSn : n P Nq is a sequence of stopping times, then supn Sn is also a stopping time, and if, moreover, the filtration pFt : t ě 0q is right continuous, then inf n Sn is stopping time as well. Proof. The proof is left as an exercise for the reader.. . 5.95. Definition. Let T be a stopping time for the filtration pFt : t ě 0q. The σ-field of events which precedes T is defined by č FT :“ tA P F8 : A X tT ď tu P Ft u . tě0. 376 Download free eBooks at bookboon.com.

(141) Advanced stochastic processes: Part II. 6. STOCHASTIC PROCESSES. Some related results 377. Indeed, the collection FT is a σ-ﬁeld and if T “ t is a ﬁxed time, then FT “ Ft . If S ď T is also a stopping time, then FS Ă FT . If the ﬁltration pFt : t ě 0q is continuous from the right, then an event A belongs to FT if and only if A belongs to F8 , and if for every t ą 0 the event A X tT ă tu belongs to Ft . If S and T are stopping times, then FminpS,T q “ FS X FT . If the ﬁltration pFt : t ě 0q times which is right continuous and if pSn : n P Nq is a sequence of stopping Ş converges downward to S, then S is a stopping time and nPN FSn “ FS .. 5.96. Definition. A process X : r0, 8q ˆ Ω Ñ E is called progressively measurable for the ﬁltration pFt : t ě 0q if for every t ą 0 the restriction of X to r0, ts ˆ Ω is measurable for the σ-ﬁelds Br0, ts b Ft and E. 5.97. Theorem. If X is right-continuous adapted process, then X is progressively measurable.. Proof. Deﬁne the sequence of processes pX n : n P Nq by means of the formula: # ` ˘ k`1 t, ω , if k2´n t ă u ď pk ` 1q2´n t, 0 ď k ď 2n ´ 1; X n 2 X n pu, ωq “ 0, if u “ 0. (5.166) Let B P E. Then we have tX n P Bu. “ t0u ˆ tXp0q P Bu Y P Br0, ts b Ft .. ď. 0ďkď2n ´1. ˆˆ. ȷ " ˆ ˙ *˙ k k`1 k`1 ˆ X PB , 2n 2n 2n (5.167). So X n is progressively measurable. Because the process X is P-almost surely right-continuous it follows that limnÑ8 X n “ X, and, consequently, X is progressively measurable. This completes the proof of Theorem 5.97. 5.98. Theorem. Suppose that X is progressively measurable for the filtration pFt : t ě 0q. Let T be a stopping time. The the state variable XpT q : ω ÞÑ XpT pωq, ωq measurable for the σ-fields E and FT . Proof. On the event tT ď tu the mapping ω ÞÑ X pT pωq, ωq is the composition of the mapping ω ÞÑ pT pωq, ωq, which goes from tT ď tu to r0, ts ˆ Ω and which is measurable for the σ-ﬁelds Ft and Br0, ts b Ft , and the mapping pu, ωq ÞÑ Xpu, ωq, which goes from r0, ts ˆ Ω to E and which is measurable for the σ-ﬁelds Br0, ts b Ft and E. (In the latter argument the progressive measurability of X was used.) The composition of measurable mappings is again measurable, and hence XpT q is measurable for de σ-ﬁelds FT and E. This completes the proof of Theorem 5.98.. . 5.99. Corollary. If T is a stopping time and if X is progressively measurable, T then the process X T defined ˘ by X puq “ XpminpT, uqq is adapted to the stopped ` filtration FminpT,uq : u ě 0 . 377 Download free eBooks at bookboon.com.

(142) Advanced stochastic processes: Part II 378. 5. SOME RELATED RESULTS. Some related results. Proof. The proof is left as an exercise for the reader.. . The next lemma is often employed instead of the monotone class theorem. 5.100. Lemma. Let F be a σ-field on Ω and H a vector space consisting of F-measurable real-valued bounded functions on Ω. Suppose that the following hypotheses are fulfilled: (1) H contains the constant functions; (2) If f and g belong to H, then the product f g belongs to H; (3) If f is the pointwise limit of a sequence of functions pfn : n P Nq in H, for which |fn | ď 1, then f belongs to H; (4) F “ σ pf : f P Hq. Then H contains all bounded F-measurable functions. Proof. Let D be the collection D “ tA P F : 1A P Hu. Then D is a Dynkin system and by (2) D is closed for taking ﬁnite intersections. So D is a σ-ﬁeld. Pick f P H and let a P R. We will prove that the set tf ě au belongs to D. By taking an appropriate combination of f and the constant function 1 we may assume that 0 ď f ď 1 and that 0 ď a ď 1. Let p be a polynomial. By (2) ppf q belongs to H. Let φ : r0, 1s Ñ R be a continuous function. By the theorem of Stone-Weierstrass there exists a sequence of polynomials ppn : n P Nq such that supxPr0,1s |φpxq ´ pn pxq| ď n´1 . Consequently, φpf q belongs to H. Since the function 1ra,8q is a (decreasing) pointwise limit of a sequence of continuous functions, it follows that 1ra,8q pf q “ 1tf ěau belongs to H. So the set tf ě au belongs to D. From which it follows that D “ F. But then we infer F Ă tA P F : 1A P Hu. From this the assertion in Lemma 5.100 immediately follows. 5.101. Definition. Let pFt : t ě 0q be a ﬁltration on the probability space pΩ, F, Pq , and let X be an adapted process. (i) The process X is called a martingale (relative to P and to the ﬁltration 1 pFt : t ě 0q) if for every t ě 0 the variable Xptq belongs ˘ F, Pq ` to Lˇ pΩ, and if for every pair 0 ď s ă t the equality Xpsq “ E Xptq ˇ Fs holds P-almost surely. (ii) The process X is called a sub-martingale (relative to P and to the ﬁltration pFt : t ě 0q) if for every t ě 0 the variable Xptq `belongs ˇ to˘ L1 pΩ, F, Pq and if for every s ă t the inequality Xpsq ď E Xptq ˇ Fs holds P-almost surely. (iii) The process X is called a super-martingale (relative to P and to the ﬁltration pFt : t ě 0q) if for every t ě 0 the variable Xptq `belongs ˇ to˘ L1 pΩ, F, Pq and if for every s ă t the inequality Xpsq ě E Xptq ˇ Fs holds P-almost surely. 378 Download free eBooks at bookboon.com.

(143) Advanced stochastic processes: Part II. Some related results. 6. STOCHASTIC PROCESSES. 379. Instead of assuming Xptq P L1 pΩ, F, Pq in (ii) it is sometimes assumed that the variable Xptq` “ maxpXptq, 0q belongs to L1 pΩ, F, Pq. In (iii) it is sometimes only assumed that Xptq´ “ max p´Xptq, 0q belongs to L1 pΩ, F, Pq. If T is a (discrete) subset of r0, 8q and if pXptq, Ft qtě0 is a martingale (sub-martingale, super-martingale), then the process pXptq, Ft qtPT is so as well. Then we can use “discrete results” and via a limiting procedure we then obtain results in the “continuous case”. 5.102. Definition. Let f : r0, 8q Ñ R be a function, let T Ď r0, 8q and let ˇ a ă b be real numbers. Deﬁne the number of upcrossings UT pf, a, bq of f ˇT between a and b by UT pf, a, bq. “ sup tm : there exist t1 ă t2 ă . . . ă t2m , tj P T f pt2k´1 q ď a, f pt2k q ě bu . (5.168). I joined MITAS because I wanted real responsibili� I joined MITAS because I wanted real responsibili�. Real work International Internationa al opportunities �ree wo work or placements. �e Graduate Programme for Engineers and Geoscientists. Maersk.com/Mitas www.discovermitas.com. �e G for Engine. Ma. Month 16 I was a construction Mo supervisor ina const I was the North Sea super advising and the No he helping foremen advis ssolve problems Real work he helping fo International Internationa al opportunities �ree wo work or placements ssolve pr. 379 Download free eBooks at bookboon.com. Click on the ad to read more.

(144) Advanced stochastic processes: Part II 380. 5. SOME RELATED RESULTS. Some related results. 5.103. Lemma. Let D by the set of non-negative dyadic numbers and let f : D Ñ R be a function, which is bounded on D X r0, ns for all n P N. Assume that, for all n P N and for all real numbers a ă b, with a and b (dyadic) rational, the number of upcrossings UDXr0,ns pf, a, bq of f is finite. Then the following assertions are true: (a) For every t P R the following left and right limits exist: lim f psq and. sÒt,sPD. lim f psq;. sÓt,sPD. (5.169). (b) Define the function g by gptq “ lim f psq. Then g is right-continuous sÓt,sPD. and for every t ą 0 the left limit limsÒt gpsq exists.. Proof. (a) We will show that the limit limsÓt,sąt f psq exists. Since the function f is bounded it suﬃces to prove that lim inf sÓt,sąt f psq “ lim supsÓt,sąt f psq. Assume that this not the case. Then there exist dyadic rational numbers a and b such that lim inf sÓt,sąt f psq ă a ă b ă lim supsÓt,sąt f psq. This means that there exists s0 ą t, s0 P D, with f ps0 q ą b. There also exists s1 ă s0 , s1 ą t, s1 P D, such that f ps1 q ă a. In general we obtain t ă s2k´1 ă s2k´2 , s2k´1 P D, for which f ps2k´1 q ă a and we obtain t ă s2k ă s2k´1 , s2k P D, with f ps2k q ą b. For m P N we write t2m “ s0 , t2m´1 “ s1 , . . ., t2 “ s2m´1 , t1 “ s2m´1 . Pick n ą s0 . Then we have UDXr0,ns pf, a, bq ě m. Since m P N is arbitrary it follows that UDXr0,ns pf, a, bq “ 8. So we obtain a contradiction. The existence of the left limit can be treated similarly. (b) Put gptq “ limsÓt,sąt,sPD f psq. By (a) this function is well deﬁned. Since, for every n P N, the function f is bounded on the set D X r0, ns the function g possesses this property as well. Let now ptn : n P Nq be a sequence that decreases to t and for which tn ą t for all n P N. We will prove limnÑ8 gptn q “ gptq. Then this shows that g is right-continuous at t. Assume lim inf nÑ8 gptn q ă gptq. This will lead to a contradiction. By passing to a subsequence, which we call again ptn : n P Nq, we may suppose that lim inf nÑ8 gptn q “ limnÑ8 gptn q and that there are numbers a and b P D such that for all n P N, gptn q ă a ă b ă gptq. Then pick s0 ą t0 such that f ps0 q ă a: this possible, because gpt0 q ă a. The pick s0 ą t0 ą s1 ą t in such a way that f ps1 q ą b: this is possible, because gptq ą b. Then choose tn2 , s1 ą tn2 ą t, with gptn2 q ă a. Then there exists s1 ą s2 ą tn2 , such that f ps2 q ă a. This is so because gptn2 q ă a. This procedure can be continued. Like in (a) we arrive at UDXr0,ns pf, a, bq “ 8, for a certain nN, n ą t. This is a contradiction. But then it follows that lim inf nÑ8 gptn q ě gptq. In the same fashion we see that lim supnÑ8 gptn q ď gptq. Consequently, gptq “ limnÑ8 gptn q. In order to prove the existence of the left limit of the function g at t, we choose a sequence ptn : n P Nq, that increases to t, and which has the property that tn ă t for all n P N. Assuming that lim inf nÑ8 gptn q ă lim supnÑ8 gptn q, then, as above, we arrive at the conclusion that, for certain dyadic numbers a ă b, for which lim inf nÑ8 gptn q ă a ă b ă lim supnÑ8 gptn q, the number of upcrossings of the function f on the interval D X r0, ns with n ą t is inﬁnite.. 380 Download free eBooks at bookboon.com.

(145) Advanced stochastic processes: Part II 6. STOCHASTIC PROCESSES. Some related results 381. This completes the proof of Lemma 5.103.. . 5.104. Theorem (Doob’s optional time theorem for sub-martingales). Let pXpjq : j P Nq. be a sub-martingale relative to the filtration pFn : n P Nq, and let T ě S be stopping times. Suppose that E r|XpT q|s ă 8 and also E r|XpSq|s ă 8. If, additionally, lim E rXpmq : T ě m ě Ss “ 0, then XpSq is measurable for the mÑ8 ˇ ‰ “ σ-field FS and the inequality E XpT q ˇ FS ě XpSq holds P-almost surely.. Proof. Let A be an event in FS . For every j, j ě 1, and for every ℓ P N, ℓ ě 0, the event A X tT ě ℓ ` ju X tS “ ℓu X A then belongs to the σ-ﬁeld Fℓ`j´1 . To see this, observe that the event tT ě ku “ Ωz tT ď k ´ 1u belongs to Fk´1 . Since pXpminpT, mqq ´ XpminpS, mqqq 1A “. m m´ℓ ÿ ÿ. ℓ“0 j“1. pXpℓ ` jq ´ Xpℓ ` j ´ 1qq 1tT ěℓ`juXtS“ℓuXA ,. it follows that E ppXpminpT, mqq ´ XpminpS, mqqq 1A q “. m m´ℓ ÿ ÿ ` ˘ E pXpℓ ` jq ´ Xpℓ ` j ´ 1qq 1tT ěℓ`juXtS“ℓuXA .. ℓ“0 j“1. Hence, E ppXpminpT, mqq ´ XpminpS, mqqq 1A q ě 0. Since, in addition, E pXpT q ´ XpSq ´ XpminpT, mqq ` XpminpS, mqqq. “ E pXpT q ´ XpSq : S ě mq ` E pXpT q ´ Xpmq : T ě m ą Sq ,. the claim in Theorem 5.104 follows.. . 5.105. Proposition. Let pXpnq : n P Nq be a (sub-)martingale relative to the discrete filtration pFn : n P Nq. (a) Let H “ pHpnq : n P N, n ě 1q be a positive bounded process with the property that Hn is measurable for the σ-field Fn´1 . Define the process pY pnq : n P Nq by Y p0q “ Xp0q,. Y pnq “ Xp0q `. n ÿ. k“1. Hpkq pXpkq ´ Xpk ´ 1qq , n ě 1.. Then the process Y is a (sub-)martingale. By putting Hpnq “ 1tnďT u , where T is a stopping time we see that process X T :“ pXpminpT, nqq : n P Nq is a (sub-)martingale.. 381 Download free eBooks at bookboon.com.

(146) Advanced stochastic processes: Part II 382 5. SOME RELATED RESULTS. Some related results. (b) Let S and T be a pair of bounded stopping times such that 0 ď S ď T . Then ˇ ˘ ` XpSq ď E XpT q ˇ FS , P-almost surely, (5.170) and if X is a martingale, then there is an equality in (5.170).. Moreover, an adapted and integrable process X is a martingale if and only if E pXpT qq “ E pXpSqq for each pair of bounded stopping times S and T for which S ď T. Proof. (a) The first assertion in (a) is easy to see. To understand the second assertion we observe that 1ř tT ěnu “ 1´1tT ďn´1u is measurable for the σ-field Fn´1 and we notice that Xp0q` nk“1 1tT ěku pXpkq ´ Xpk ´ 1qq “ XpminpT, nqq. This proves assertion (a) in Proposition 5.105. ˇ ˘ ` (b) De inequality XpSq ď E XpT q ˇ FS , P-almost surely was already proved in Theorem 5.104 and can be obtained from (a) by putting Hpnq “ 1tT ěnu ´1tSěnu . If we use the equality EpXpS B qq “ EpXpT B qq for de times S B “ S1B ` M 1B c and T B “ T 1B ` M 1B c , where B belongs to FS and where M ě T ě S, then we get E pXpT q1B ` XpM q1B c q “ E pXpSq1B ` XpM q1B c q . But, then `it follows ˇ that ˘ E pXpT q1B q “ E pXpSq1B q for all B P FS and hence XpSq “ E XpT q ˇ FS .. The proof of Proposition 5.105 is now complete.. 382 Download free eBooks at bookboon.com. . Click on the ad to read more.

(147) Advanced stochastic processes: Part II 6. STOCHASTIC PROCESSES. Some related results 383. 5.106. Theorem (Doob-Meyer decomposition for discrete sub-martingales). Let pXpjq : j P Nq be a sub-martingale. Then there exists a unique martingale M “ pM pkq : k P Nq together with a unique predictable increasing process A “ pApkq : k P Nq, with Ap0q “ 0, such that Xpkq “ M pkq ` Apkq, for k P N. 5.107. Remark. This theorem is, in an appropriate form, also true for submartingales X of de form X “ pXptq : t ě 0q (continuous time). A process A “ pApkq : k P Nq is called predictable, if Apkq is measurable for Fk´1 , and this for every k P N. Proof. Existence Define the process A by Ap0q “ 0 and Apkq “. k ÿ. j“1. ˇ ` ˘ E Xpjq ´ Xpj ´ 1q ˇ Fj´1 .. Define the process M by M pkq “ Xpkq ´ Apkq. Then the process M is a martingale and the process A is increasing (i.e. non-decreasing) and predictable. Moreover, the equality X “ M ` A holds.. Uniqueness Let the process X be such that X “ M `A where M is a martingale and where A is predictable and increasing. In addition, suppose that Ap0q “ 0. Then the equalities k ÿ. j“1. “ “. ˇ ` ˘ E Xpjq ´ Xpj ´ 1q ˇ Fj´1. k ÿ. j“1 k ÿ. j“1. k ˇ ˇ ` ` ˘ ÿ ˘ E Apjq ´ Apj ´ 1q ˇ Fj´1 E M pjq ´ M pj ´ 1q ˇ Fj´1 ` j“1. pApjq ´ Apj ´ 1qq “ Apkq,. hold for k ě 1. So the proof of Theorem 5.106 is complete now.. . 5.108. Theorem. Let X “ pXpkq : 1 ď k ď N q be a sub-martingale. Then the following inequality holds: ˘ E rmaxpXpN q ´ a, 0qs ` . E Ut1,...,N u pX, a, bq ď b´a Proof. For a proof we refer the reader to Proposition 3.71 of Chapter 3. Notice that, with X the process maxpX ´ a, 0q is also a sub-martingale. 5.109. Theorem. Let X “ pXptq : t ě 0q be a sub-martingale for the filtration pFt : t ě 0q. For a ă b the inequality. holds.. ` ˘ E rmax pXpN q ´ a, 0qs E UDXr0,N s pX, a, bq ď . b´a. 383 Download free eBooks at bookboon.com.

(148) Advanced stochastic processes: Part II 384. Some related results. 5. SOME RELATED RESULTS. Z and define Un by Un “ UDn Xr0,N s pX, a, bq. The 2n sequence Un then increases to UDXr0,N s pX, a, bq. So it follows that ` ˘ E rmaxpXpN q ´ a, 0qs . E UDXr0,ns “ lim E pUn q ď nÑ8 b´a This completes the proof of Theorem 5.109. Proof. Write Dn “. The following theorem contains Doob’s maximal inequalities for submartingales. 5.110. Theorem. Let X “ pXp0q, . . . , Xpnqq be a sub-martingale. Then the following maximal inequalities of Doob hold: ˙ ˆ 1 (a) P max Xj ě λ ď EpXpnqq; 0ďjďn λ ˆ ˙ 2 P max |Xj | ě λ ď E p5 |Xpnq| ´ 2Xp0qq ; (b) 0ďjďn λ and if X is a martingale ˆ ˙ 1 P max |Xj | ě λ ď tE p|Xpnq|qu . 0ďjďn λ. (c). Proof. We begin with a proof of (c). Consider the mutually disjoint events " * A0 “ t|X0 | ą λu , and Ak :“ |Xpkq| ą λ, max |Xpjq| ď λ , 0ďjďk´1 Ťn 1 ď k ď n. Then k“0 Ak “ tmax0ďjďn |Xpjq| ě λu. Therefore ȷ ÿ „ n P pAj q , P max |Xpjq| ě λ “ 0ďjďn. j“0. and so, using the martingale property 1 PpAk q “ E p1Ak q ď E r1Ak |Xpkq|s λ (martingale property) “. ˇ ˘‰ 1 ˇ ‰ 1 “ ` 1 “ E 1Ak |Xpnq| ˇ Fk ď E 1Ak E |Xpnq| ˇ Fk “ E r1Ak |Xpnq|s . λ λ λ (5.171). By summing over k in (5.171)we get (c).. (a) The proof of (a) follows almost the same lines, except that in the definitions of the events Ak the absolute value signs have to be omitted. (b) For the proof of this assertion we employ the Doob-Meyer decomposition theorem (Theorem 5.106). Write X “ M ` A with M a martingale, and A (predictable) increasing process. We let M p0q “ Xp0q. Then we see ˙ ˆ ˙ ˙ ˆ ˆ λ λ ` P Apnq ě P max |Xpjq| ě λ ď P max |M pjq| ě 0ďjďn 0ďjďn 2 2. 384 Download free eBooks at bookboon.com.

(149) Advanced stochastic processes: Part II. 6. STOCHASTIC PROCESSES. Some related results 385. (by (c)) 2 2 E |M pnq| ` EpAn q λ λ 2 ď E p|M pnq| ´ M pnq ` Xpnqq λ 2 ď E p2 |Xpnq| ` 2Apnq ` Xpnqq λ 2 ď E p5 |Xpnq| ´ 2Xp0qq . λ This proves assertion (b). ď. . The proof of Theorem 5.110 is complete now.. 5.111. Lemma. Let pAn : n P Nq be a sequence of Ş σ-fields decreasing to the σfield A8 . So that An`1 Ď An , n P N, and A8 “ nPN An . Let pfn : n P Nq Y tf8 u be a sequence of stochastic variables with the following properties: (i) fn is An`-measurable, n P N, and f8 is A8 -measurable; ˇ ` ˇ ˘ ˘ ˇ (ii) fm ď E fn Am , for all m ě n, and f8 ď E fn ˇ A8 , n P N; (iii) limnÑ8 E pfn q “ E pf8 q.. Then the sequence pfn : n P Nq is uniformly integrable.. Proof. For m “ 1, 2, . . . , 8 we have ˇ ` ˇ ` ˘ ˘ fm ď E f1 ˇ Am and maxpfm , 0q ď E maxpfm , 0q ˇ Am .. From this it follows that the sequence pmaxpfm , 0q : 1 ď m ď 8q is dominated by an integrable function (in fact by E pmaxpf1 , 0qq). So it follows that this sequence is uniformly integrable. The fact that the sequence pmaxp´fn , 0q : n P Nq is also uniformly integrable, is much less trivial. To this end we consider ´ λP pfn ă ´λq ě E pfn : fn ă ´λq “ E pfn q ´ E pfn : fn ě ´λq ˘ ` ` ˇ ˘˘ ` ` ˇ ˘ ě E E fn ˇ A8 ´ E E f1 ˇ An : fn ě ´λ ě E pf8 q ´ E pf1 : fn ě ´λq ě E pf8 q ´ E pmaxpf1 , 0qq .. (5.172). From (5.172) it follows that λP pfn ă ´λq ď E pmaxpf1 , 0q ` maxp´f8 , 0qq ă 8.. Then choose ϵ ą 0 and m0 in such a way that E pfm0 q ď E pf8 q ` ϵ. For n ě m0 we then see E pfm0 q ď E pfn q ` ϵ. Hence, E pfn q ě E pfm0 q ´ ϵ. Then choose δ ą 0 such that PpAq ď δ implies E p|fk | : Aq ď ϵ for k “ 1, . . . , m0 . After that choose λ0 so large that P pfk ă λq ď δ for all k P N and for all λ ě λ0 . For 1 ď k ď m0 we then get E p|fk | : fk ď ´λq ď ϵ, λ ě λ0 . For k ě m0 we see. E pfk : fk ă ´λq “ E pfk q ´ E pfk : fk ě λq (5.173) ˇ ˘ ˘ ` ` ě E pfm0 q ´ E E fm0 ˇ Ak : fk ě ´λ ´ ϵ ě E pfm0 : fk ă ´λq ´ ϵ. 385 Download free eBooks at bookboon.com.

(150) Advanced stochastic processes: Part II 386 5. SOME RELATED RESULTS. Some related results. By (5.173) we obtain E p|fk | : fk ă ´λq “ ´E pfk : fk ă ´λq ď |E pfm0 : fk ă ´λq| ` ϵ ď 2ϵ. for a certain λ ą 0. Thus we see that the sequence pmaxp´fn , 0q : n P Nq is also uniformly integrable. This yields the desired result in Lemma 5.111. 5.112. Theorem. Let, relative to the right-continuous filtration pFt : t ě 0q, the process X be a sub-martingale. Suppose that F0 contains the zero-sets, and that the function t ÞÑ E pXptqq is right-continuous. Then there exists a process Y “ pY ptq : t ě 0q which is cadlag is which cannot be distinguished from X. So for every t ě 0 the equality Y ptq “ Xptq holds P-almost surely. Proof. There exists an event Ω1 in Ω, with PpΩ1 q “ 1, such that on Ω1 the following claims hold: sup tPDXr0,ns. |Xptq| ă 8, for all n P N;. UDXr0,ns pX, a, bq ă 8, or all n P N and for all a ă b, a and b rational.. Since P pΩ1 q “ 1 we see that Ω1 belongs to F0 and, hence Ω1 belongs to Ft for all t ě 0. On Ω1 we deﬁne the process Y “ pY ptq; t ě 0q as follows: Y ptq “ limsÓt,sąt,sPD Xpsq. Then Y ptq is measurable for all σ-ﬁelds Fu with u ą t. By the right continuity of the ﬁltration pFt : t ě 0q we then see that Y ptq is measurable for the σ-ﬁeld Ft . Then take t “ limnÑ8 sn , where sn Ó t, and where, for every n P N, sn belongs to D. Then Y ptq “ limnÑ8 Xpsn q in probability. Then apply Lemma 5.111 to conclude that the sequence pXpsn q : n P Nq is uniformly integrable, and hence Y ptq “ L1 ´ limnÑ8 Xpsn q. We may apply Lemma 5.111. for fn :“` Xpsn q,ˇ f8˘ “ Y ptq, A8 “ Ft and An “ Fsn . Then notice that 1 Xptq ď E Xpsn q ˇ F`t , P-almost ˇ ˘ surely. By L -convergence, from the latter we see that Xptq ď E Y ptq ˇ Ft and thus Xptq ď Y ptq P-almost surely. Since, in addition, E pY ptqq “ limnÑ8 EpXpsn qq “ EpXptqq, the equality Y ptq “ Xptq follows P-almost surely. This completes the proof of Theorem 5.112.. . 5.113.“Theorem. Let X “ pXptq : t ě 0q be a sub-martingale with property that ‰ sup E Xptq` ă 8. The following assertions hold true. tě0. (a) The limit Xp8q :“ limsÑ8,sPD Xpsq exists P-almost surely. (b) If X is a cadlag process, then the limit Xp8q :“ limsÑ8 exists P-almost surely. ` (c) If, in addition, the process ` pXptqˇ :˘t ě 0q is uniformly integrable, then the inequality Xptq ď E Xp8q ˇ Ft holds.. Proof. (a) From the maximal inequality of Doob it follows that, for λ ą 0, the following inequality holds: ¸ ˜ ` ˘ (5.174) λP sup |Xptq| ą λ ď 10E Xpnq` ` Xp0q´ . tPDXr0,ns. 386 Download free eBooks at bookboon.com.

(151) Advanced stochastic processes: Part II 6. STOCHASTIC PROCESSES. Some related results 387. By letting n tend to 8 in (5.174) we obtain ˙ ˆ ` ˘ λP sup |Xptq| ą λ ď 10 sup E Xpnq` ` Xp0q´ , tPD. n. and hence, suptPD |Xptq| ă 8 P-almost surely. In the same manner we see ˘ ` E pXpnq ´ aq` . (5.175) E UDXr0,8q pX, a, bq ď sup b´a n From (5.175) we see that UDXr0,8q pX, a, bq ă 8 P-almost surely. As we proved regularity starting from (5.175) and (5.174) (in fact from their consequences), we now obtain that Xp8q :“ limsÑ8,sPD Xpsq exists. (b) If X is cadlag, then, like in the proof of the regularity, the limit Xp8q “ limsÑ8 Xpsq exists.. no.1. Sw. ed. en. nine years in a row. STUDY AT A TOP RANKED INTERNATIONAL BUSINESS SCHOOL Reach your full potential at the Stockholm School of Economics, in one of the most innovative cities in the world. The School is ranked by the Financial Times as the number one business school in the Nordic and Baltic countries.. Stockholm. Visit us at www.hhs.se. 387 Download free eBooks at bookboon.com. Click on the ad to read more.

(152) Advanced stochastic processes: Part II 388 5. SOME RELATED RESULTS. Some related results. (c) Since the process pXptq` : t ě 0q is uniformly integrable, it also follows that the process t ÞÑ maxpXptq, aq “ pXptq ´ aq` ` a is uniformly integrable as well. So, for A P Ft and for u ą t, the (in-)equalities ż ż maxpXptq, aq dP ď lim maxpXpuq, aq dP uÑ8 A A ż ż “ lim maxpXpuq, aq dP “ maxpXp8q, aq dP A uÑ8. hold true. Since. ż. A. `. Xp8q dP “ lim. uÑ8. ż. Xpuq` dP ă 8. we see that Xp8q` belongs to L1 pΩ, F, Pq. But then we get ż ż ż Xptq dP “ lim maxpXptq, aq dP ď lim maxpXp8q, aq dP aÑ´8 A aÑ´8 A A ż “ Xp8q dP. (5.176) A ˇ ˘ ` From (5.176) the inequality Xptq ď E Xp8q ˇ Ft follows. This proves item (c). The proof of Theorem 5.113 is now complete. . 5.114. Theorem. Let X “ pXptq : t ě 0q be a sub-martingale with the property that the process pXptq` : t ě 0q is uniformly integrable. In addition, suppose that X is cadlag.If S and T are a pair of stopping `times such ˇ ˘that 0 ď S ď T ď 8, then the following inequality holds: XpSq ď E XpT q ˇ FS .. Proof. Put Sn “ 2´n r2n T s and, similarly, Tn “ 2´n r2n T s. Then the stopping times Sn and Tn attain exclusively discrete values (in fact they take their values in 2´n N). It is true that Sn Ó S (if n Ñ 8) and the same is true for the sequence pTn : n P Nq. Moreover, Sn ď Tm for n ě m. From Doob’s theorem about discrete optional stopping times it follows that ˇ ˇ ` ` ˘ ˘ XpSn q ď E XpTm q ˇ FSn , XpSn q ď E Xp8q ˇ FSn , ˇ ` ˘ XpSn q ď E Xp8q ˇ FSn . From this it follows that the processes pXpSn q` : n P Nq and pXpTn q` : n P Nq are uniformly integrable. For all n, m in N, n ě m, the following inequality holds for A P FS : ż ż maxpXpSn q, aq dP ď maxpXpTm q, aq dP; (5.177) A. A. (let n tend to 8 in (5.177) to obtain) ż ż maxpXpSq, aq dP ď maxpXpTm q, aq dP; A. (in (5.178) let m tend to 8 to obtain) ż ż maxpXpSq, aq dP ď maxpXpT q, aq dP; A. (5.178). A. A. 388 Download free eBooks at bookboon.com. (5.179).

(153) Advanced stochastic processes: Part II. 6. STOCHASTIC PROCESSES. (in (5.179) let a tend to ´8 to obtain) ż ż XpSq dP ď XpT q dP, A. Some related results 389. (5.180). A. and that limnÑ8 XpSn q “ XpSq and that ˘ is true for the stopping time ` theˇ same T . By (5.180) we then see XpSq ď E XpT q ˇ FS . This completes the proof of Theorem 5.114. 5.115. Corollary. Let X “ pXpsq : 0 ď s ď tq be a cadlag martingale and let 0 ď S ď T ď t be two stopping times. The following equalities are true: ˇ ˘ ` XpSq “ E XpT q ˇ FS and E pXpT qq “ E pXp8qq “ EpXp0qq.. Proof. The proof is left as an exercise for the reader. Among other things notice that the martingale pXpsq : 0 ď s ď tq is uniformly integrable. . 5.116. Corollary. Let X be a cadlag martingale in L1 pΩ, F, Pq which is uniformly integrable. Then the limit Xp8q :“ limtÑ8 Xptq exists P-almost surely, and if S and T are stopping times such that 0 ď S ď T ď 8, then the following equalities hold: ˇ ˘ ` XpSq “ E XpT q ˇ FS and E pXpT qq “ E pXp8qq “ EpXp0qq.. Proof. The proof of this corollary is left as an exercise for the reader. Observe that for n P N ﬁxed the martingale pXpminpn, tqq : t ě 0q is uniformly integrable. . In what follows the process X : r0, 8q ˆ Ω Ñ Rν is a process with values in Rν , where ν may be 1. 5.117. Definition. Let X be a stochastic process, which is adapted to the ﬁltration pFt : t ě 0q. The process X is said to be a L´evy process if X possesses the following properties: (a) For all s ă t the variable Xptq ´ Xpsq is independent of Fs ; (b) For all s ď t the variable Xptq ´ Xpsq has the same distribution as Xpt ´ sq; (c) For all t ě 0 and for every sequence ptn : n P Nq in r0, 8q that converges to t, the limit limnÑ8 Xptn q “ X exists in P-law (or in P-measure). Sometimes this is denoted by P-limnÑ8 Xptn q “ Xptq. 5.118. Theorem. Let X be a stochastic process, which is adapted to the filtration pFt : t ě 0q, and which takes it values in Rν . The following assertions are true: (a) Let X be a L´evy-process. Define for t ě 0 the probability measure µt as being the distribution of Xptq. So µt pBq “ P pXptq P Bq, where B is a Borel subset of Rν . Then the family tµt : t ě 0u is a vaguely continuous semigroup of probability measures. (b) Conversely, let tµt : t ě 0u be a vaguely continuous semigroup of probability measures on Rν . Then there exists a L´evy-process X “ tXptq : t ě 0u. 389 Download free eBooks at bookboon.com.

(154) Advanced stochastic processes: Part II 390 5. SOME RELATED RESULTS. Some related results. with cadlag paths such that µt pBq “ P pXptq P Bq for all Borel subsets B of Rν . Proof. (a) Deﬁne for t ě 0 the characteristic function ft of Xptq as being the Fourier transform of the P-distribution of Xptq. So that ft pξq “ E pexp p´i ⟨ξ, Xptq⟩qq , ξ P Rν . Since, for t, s P r0, 8q, Xps ` tq “ Xps ` tq ´ Xpsq ` Xpsq, since, in addition, Xps ` tq ´ Xpsq is independent of Xptq, and because Xps ` tq ´ Xpsq possesses the same distribution as Xptq we infer fs`t pξq “ E pexp p´i ⟨ξ, Xps ` tq⟩qq. “ E pexp p´i ⟨ξ, Xps ` tq ´ Xpsq⟩q exp p´i ⟨ξ, Xpsq⟩qq. “ E pexp p´i ⟨ξ, Xps ` tq ´ Xpsq⟩qq E pexp p´i ⟨ξ, Xpsq⟩qq “ E pexp p´i ⟨ξ, Xptq⟩qq E pexp p´i ⟨ξ, Xpsq⟩qq “ ft pξqfs pξq.. (5.181). Since Xp0q and Xp0q ´ Xp0q “ 0 have the same distribution we see f0 pξq “ 1. Since P-limuÓ0 Xpuq “ Xp0q we see, for example by Theorem 5.85 in combination with the implication (1) ñ (9) of Theorem 5.43, that limsÓ0 fs pξq “ f0 pξq “ 1. From (5.181) it then follows that lim ft pξq ´ fs pξq “ lim fs pξq pft´s pξq ´ f0 pξqq “ 0 tÓs. tÓs. for all s ě 0. Because, by applying equality (5.181) repeatedly, we see ft pξq “ n pft2´n pξqq2 . In addition we have limsÓ0 fs pξq “ 1. So it follows that for no value of t P r0, 8q the function ft pξq vanishes for any ξ. Since, for t ă s, ft pξq´fs pξq “ pf0 pξq ´ fs´t pξqq ft pξq, it also follows that limtÒs ft pξq “ fs pξq, for s ą 0. From the previous considerations it follows that the function t ÞÑ ft pξq, t P r0, 8q, is a continuous function, which satisﬁes the relation fs`t pξq “ fs pξqft pξq for all s, t ě 0 and this for all ξ P Rν . Furthermore, we deﬁne the family of measures tµt : t ě 0u as being the P-distributions of the L´evy-process X. So that µt pBq “ P pXptq P Bq, B Borel subset of Rν . From the previous arguments it then follows that ps pξqp ps`t pξq “ fs`t pξq “ fs pξqft pξq “ µ µt pξq µ. (5.182). ps pξq “ 1. So that the family tµt : t ě 0u is a vaguely continuous and that limsÓ0 µ semigroup of probability measures on Rν . By Theorem 5.31 there then exists a pt pξq “ exp p´tψpξqq. continuous negative-deﬁnite function ψ such that ft pξq “ µ. (b) Deﬁne pΩ, F, Pq as in Proposition 5.36. Likewise we deﬁne the state variables Xptq : Ω Ñ Rν as in Proposition 5.36. Let the ﬁltration pFt : t ě 0q be determined by Ft “ σ pXpuq : 0 ď u ď tq. So the ﬁltration pFt : t ě 0q is the internal history of the process X. Then X is a L´evy-process, which possesses the properties as described in (b). The fact that for t ą s the variable Xptq ´ Xpsq is independent of Fs was proved in Theorem 5.37. We must show. 390 Download free eBooks at bookboon.com.

(155) Advanced stochastic processes: Part II 6. STOCHASTIC PROCESSES. Some related results 391. that, for ϵ ą 0 fixed, limsÓ0 P p|Xpsq ´ Xp0q| ą ϵq “ 0. Therefore, notice first that P pXp0q “ 0q “ µ0 t0u “ 1. Hence, with Bpϵq “ tx P Rν : |x| ď ϵu, we have P p|Xpsq ´ Xp0q| ą ϵq “ P p|Xpsq ´ Xp0q| ą ϵ, Xp0q “ 0q “ P p|Xpsq| ą ϵ, Xp0q “ 0q. “ P p|Xpsq| ą ϵq. “ µs tRν zBpϵqu “ 1 ´ µs tBpϵqu .. (5.183). Since the convolution semigroup tµt : t ě 0u is vaguely continuous it follows that lim µs tBpϵqu “ 1. From (5.183) we then see that limsÓ0 P p|Xpsq ´ Xp0q| ą ϵq “ sÓ0. 0. The only problem which is still left, is the fact that the process X is not necessarily cadlag. In the following propositions and lemmas we will, among other things, resolve this problem. From Theorem 5.121 it follows that the process X is also a L´evy process for the filtration pGt : t ě 0q, where Gt “ Ft YN. By Theorem 5.123 we then see that the process X possesses a cadlag version. The proof of Theorem 5.118 is now complete.. 391 Download free eBooks at bookboon.com. . Click on the ad to read more.

(156) Advanced stochastic processes: Part II 392 5. SOME RELATED RESULTS. Some related results. 5.119. Proposition. Suppose 0 ď s1 ă ¨ ¨ ¨ ă sm and choose t ě 0. Let X be a L´evy process for the filtration pFt : t ě 0q, where Ft “ σ tXpuq : 0 ď u ď tu. Let tµt : t ě 0u be the corresponding convolution semigroup and ψ the corresponding negative-definite function. So µt pBq “ P pXptq P Bq for all Borel subsets B and µ pt pξq “ exp p´tψpξqq for all t ě 0. For ξ 1 , . . . , ξ m in Rν the following equalities hold: ¸ « ˜ ﬀ m ÿ ⟩ ˇ ⟨ j ˇ Ft E exp ´i ξ , Xpt ` sj q «. j“1. ˜. “ E exp ´i ˜. ⟨. “ exp ´i. m ÿ ⟨. j“1. m ÿ. ξ j , Xpt ` sj q ⟩¸. ξ j , Xptq. j“1. Here we write Ft` “. ⟩. Ş. sąt. ¸. ˜. ˇ ˇ Ft`. exp ´. m ÿ. j“1. ﬀ. psj ´ sj´1 q ψ. ˜. m ÿ. k“j. ξk. ¸¸. .. (5.184). Fs and s0 “ 0.. Proof. We apply induction with respect to m. We begin with the conditioning on Ft . For m “ 1 we have “ ` ⟨ ⟩˘ ˇ ‰ E exp ´i ξ 1 , X pt ` s1 q ˇ Ft ` ⟨ “ ` ⟨ ⟩˘ ˇ ‰ ⟩˘ “ E exp ´i ξ 1 , X pt ` s1 q ´ Xptq ˇ Ft exp ´i ξ 1 , Xptq (Xpt ` s1 q ´ Xptq does not depend on Ft ) “ ` ⟨ ` ⟨ ⟩˘‰ ⟩˘ “ E exp ´i ξ 1 , X pt ` s1 q ´ Xptq exp ´i ξ 1 , Xptq. (Xpt ` s1 q ´ Xptq has the same distribution as Xps1 q) “ ` ⟨ ` ⟨ ⟩˘‰ ⟩˘ “ E exp ´i ξ 1 , X ps1 q exp ´i ξ 1 , Xptq ` ˘ ` ⟨ ⟩˘ “µ ps1 ξ 1 exp ´i ξ 1 , Xptq ` ˘˘ ` ` ⟨ ⟩˘ “ exp ´s1 ψ ξ 1 exp ´i ξ 1 , Xptq ` ˘˘ ` ` ⟨ ⟩˘ “ exp ´ ps1 ´ s0 q ψ ξ 1 exp ´i ξ 1 , Xptq .. (5.185). Notice that (5.185) is the same as the equality in (5.184) for m “ 1. Suppose now that we already know (5.184) for every t ě 0, for every m-tuple s1 ă ¨ ¨ ¨ ă sm and for every m-tuple ξ 1 , . . . , ξ m in Rν . We keep working with the original filtration pFt : t ě 0q. For sm`1 ą sm and for ξ m`1 P Rν we then see « ˜ ¸ ﬀ m`1 ÿ⟨ ˇ ⟩ ˇ Ft E exp ´i ξ j , X pt ` sj q «. ˜. j“1. “ E exp ´i. m ÿ ⟨. j“1. j. ⟩. ξ , X pt ` sj q. ¸. ` ⟨ ‰ˇ ⟩˘ ˇ E exp ´i ξ m`1 , X pt ` sm`1 q ˇ Ft`sm ˇ Ft “. ﬀ. 392 Download free eBooks at bookboon.com.

(157) Advanced stochastic processes: Part II 6. STOCHASTIC PROCESSES. (employ (5.185) for t ` sm`1 instead of t) ¸ « ˜ m ÿ ⟩ ⟨ j “ E exp ´i ξ , X pt ` sj q. Some related results 393. j“1. `. ⟨. ⟩˘. exp ´i ξ m`1 , X pt ` sm`1 q. ˘˘ ˇ exp ´ psm`1 ´ sm q ψ ξ m`1 ˇ Ft `. `. (induction hypothesis) ˜ ˜ ¸¸ ˜ ⟨ ⟩¸ m`1 m`1 m`1 ÿ ÿ ÿ ξ j , Xptq exp ´ psj ´ sj´1 q ψ ξk . “ exp ´i j“1. j“1. ﬀ. (5.186). k“j. But (5.186) is the same as (5.184) with m replaced by m ` 1. Next we look at the situation for the filtration tFt` : t ě 0u which is closed from the right. Without loss of generality we may assume that s1 ą 0. In case s1 “ 0 we have indeed ¸ ﬀ « ˜ m ÿ ⟩ ˇ ⟨ j ˇ Ft` E exp ´i ξ , X pt ` sj q j“1. `. «. ⟩˘. ⟨. ˜. E exp ´i. “ exp ´i ξ 1 , Xptq. m ÿ ⟨. j“2. ⟩. ξ j , X pt ` sj q. ¸. ﬀ ˇ ˇ Ft` .. So assume that s1 ą 0 and choose n P N such that s1 ą n´1 . Then we see, by (5.184) for t ` n´1 instead of t, ¸ « ˜ ﬀ m ÿ ⟩ ˇ ⟨ j ˇ Ft` E exp ´i (5.187) ξ , X pt ` sj q « «. j“1. ˜. “ E E exp ´i. m ÿ ⟨. j“1. ⟩. ξ j , X pt ` sj q. ¸. (write s0 “ n´1 in what follows) « ˜ ⟨ ⟩¸ m ÿ ` ˘ ξ j , X t ` n´1 “ E exp ´i ˜. j“1. exp ´. m ÿ. j“1. psj ´ sj´1 q ψ. ˜. m ÿ. k“j. ﬀ ﬀ ˇ ˇ ˇ Ft`n´1 ˇ Ft`. ξk. ¸¸. In (5.187) we let n tend to 8. Apparently it follows that « ˜ ¸ ﬀ m ÿ ⟩ ˇ ⟨ j ˇ Ft` E exp ´i ξ , X pt ` sj q «. j“1. ˜. “ E exp ´i. ⟨. m ÿ. j“1. ⟩¸. ξ j , Xptq. ˜. exp ´. m ÿ. j“1. ﬀ ˇ ˇ Ft` .. psj ´ sj´1 q ψ. ˜. 393 Download free eBooks at bookboon.com. m ÿ. k“j. ξk. ¸¸. ˇ ˇ Ft`. ﬀ.

(158) Advanced stochastic processes: Part II 394. Some related results. 5. SOME RELATED RESULTS. ˜. “ exp ´i. ⟨. m ÿ. j“1. ξ j , Xptq. ⟩¸. ˜. exp ´. m ÿ. j“1. psj ´ sj´1 q ψ. ˜. m ÿ. k“j. ξk. ¸¸. .. (5.188). From (5.188) it then follows that (5.184) holds for the ﬁltration tFt` : t ě 0u which is closed from the right. This completes the proof of Proposition 5.119.. . 5.120. Corollary. Let the assumptions and hypotheses be as in Proposition 5.119. For every bounded complex-valued random variable Y , that is measurable for the σ-field σ tXpuq : u ě 0u the following equality holds P-almost surely de equality: “ ˇ ‰ “ ˇ ‰ E Y ˇ Ft “ E Y ˇ Ft` . (5.189). ¯ Proof. Put Y “ exp ´i j“1 ⟨ξ , Xpsj q⟩ . By Proposition 5.119 we see that for all such random variables Y the equality in (5.189) holds, provided that sj ě t, for 1 ď j ď m. By splitting and using the standard properties of a conditional expectation we see that the restriction sj ě t is superﬂuous. In other ´ words the equality in ¯ (5.189) holds for all variables Y of the form řm j Y “ exp ´i j“1 ⟨ξ , Xpsj q⟩ where all sj belong to r0, 8q and where all ξ j are members of Rν . Let Y0 be a bounded complex-valued random ¯ ´ ř variable, which m j belongs to the linear span of variables of the form exp ´i j“1 ⟨ξ , Xpsj q⟩ . Then consider the vector space HpY0 q deﬁned by ´. řm. j. HpY0 q. “ tY : Ω Ñ C : Y is bounded and measurable for the σ-ﬁeld σ tXpuq : u ě 0u ˇ ˘ ˇ ` ˘( ` and E Y Y0 ˇ Ft “ E Y Y0 ˇ Ft` .. By employing Lemma 5.100 or, even better, the monotone class theorem we see that HpY0 q contains all complex-valued bounded random variables, which are measurable for the σ-ﬁeld σ tXpuq : u ě 0u. Among others we may put Y0 “ 1, and the claim in Corollary 5.120 follows. 5.121. Theorem. Let X “ tXptq : t ě 0u be a L´evy-process. Let H “ tHt : t ě 0u be the internal history of the process X. Let N be the null sets in H8 . Then the filtration G, with Gt “ σ tHt Y Nu, is continuous from the right. ` ˇ ˘ Proof. Let A P Gt` . By`Corollary 5.120 we have 1 “ E 1A ˇ Ft . Let A ˇ ˘ B P Ft be such that 1B “ E 1A ˇ Ft , P-almost surely. Then P pA△Bq “ 0. Since A “ B△ pA△Bq, we see that A in fact belongs to Gt .. 5.122. Lemma. Let pxn : n P Nq be a sequence of vectors in Rν with the property that the sequence pexp p´i ⟨ξ, xn ⟩q : n P Nq converges for almost all ξ P Rν . Then the sequence pxn : n P Nq converges.. Proof. Fix 1 ď j ď ν, and let U j be a vector valued stochastic variable which is zero for the coordinates k “ j and with the property that Ujj is uniformly distributed on the interval r0, 1s. The following inequalities are true for. 394 Download free eBooks at bookboon.com.

(159) Advanced stochastic processes: Part II 6. STOCHASTIC PROCESSES. Some related results 395. 0 ă δ ă 1:. ˇ ` ` ⟨ ` ⟨ ⟩˘˘ ⟩˘ˇ2 2Re E 1 ´ exp ´i U j , xn ´ xm “ E ˇ1 ´ exp ´i U j , xn ´ xm ˇ ˙ * ˆ " ⟨ j ⟨ j ⟩ ⟩ 2 1 2 2 1 2 ě 4δ P sin U , xn ´ x m U , xn ´ xm ě δ “ 4E sin 2 2 #ˆ ˙ + ˙ ˆ 2 2 ⟩ ⟨ 2 π 1 2 ě 4δ 2 P ě δ2 min U j , xn ´ xm , π 4 4 ␣ˇ⟨ ( ⟩ˇ ě 4δ 2 P ˇ U j , xn ´ xm ˇ ě δπ ˙ ˆ δπ 2 “ 4δ max 1 ´ ,0 . |xn,j ´ xm,j | Since this inequality holds for every 0 ă δ ă 1, it follows that the j-th coordinate pxn,j : n P Nq of the sequence pxn : n P Nq converges. This holds for 1 ď j ď ν. Hence, the limit limnÑ8 xn exists. This makes the proof of Lemma 5.122 complete. Among other things, in Theorem 5.123 the proof of item (b) in 5.118 is completed. 5.123. Theorem. Let pXptq, Ft qtě0 be a L´evy-process. Suppose that the filtration pFt : t ě 0q is right-continuous, and that F0 contains the null sets. Then there exists a cadlag modification of X “ pXptq : t ě 0q.. 395 Download free eBooks at bookboon.com. Click on the ad to read more.

(160) Advanced stochastic processes: Part II 396 5. SOME RELATED RESULTS. Some related results. The result in Theorem 5.123 can be applied in case we take the internal history, completed with null sets, as the ﬁltration pFt qtě0 . Proof of Theorem 5.123. We will make use of the following product set: ) ! E “ CQ` “ pαt qtPQ` : αt P C for all t P Q` ,. endowed with the product-σ-ﬁeld E “ btPQ` BpCq. Put ) ! D “ pαt qtPQ` : Dφ : r0, 8q Ñ C cadlag with φptq “ αt for all t P Q` . Upon writing φ as the pointwise limit φptq “ limnÑ8 φn ptq, where φn ptq “. 8 ÿ. k“0. ` ˘ φ pk ` 1q2´n 1rk2´n ,pk`1q2´n q ptq,. it can be proved that D belongs to the σ-ﬁeld E. Consider the mapping: Φ : Rν ˆ Ω Ñ E deﬁned by Φpξ, ωq “ exp p´i ⟨ξ, Xptq⟩q “: αt .. (5.190). The mapping Φ is measurable for the σ-ﬁelds F8 and E. As a consequence Λ :“ Φ´1 pDq belongs to the σ-ﬁeld BpRν qbF8 . So for every pair pξ, ωq P Rν ˆΩ there exists a cadlag function f : r0, 8q Ñ C with the property that the equality f ptq “ exp p´i ⟨ξ, Xptq⟩q holds for all t P Q` . Now let the negative-deﬁnite function corresponding to the process X be given by ψ. Then the process t ÞÑ exp p´i ⟨ξ, Xptq⟩ ` tψpξqq is a martingale. This is so, because, for 0 ď s ă t, we have the following equalities: ˇ ˘ ` E exp p´i ⟨ξ, Xptq⟩ ` tψpξqq ˇ Fs ˇ ˘ ` “ E exp p´i ⟨ξ, Xptq ´ Xpsq⟩ ` pt ´ sqψpξqq ˇ Fs exp p´i ⟨ξ, Xpsq⟩ ` sψpξqq (Xptq ´ Xpsq does not depend on Fs ). “ E pexp p´i ⟨ξ, Xptq ´ Xpsq⟩ ` pt ´ sqψpξqqq exp p´i ⟨ξ, Xpsq⟩ ` sψpξqq (Xptq ´ Xpsq heeft dezelfde distribution als Xpt ´ sq) “ E pexp p´i ⟨ξ, Xpt ´ sq⟩ ` pt ´ sqψpξqqq exp p´i ⟨ξ, Xpsq⟩ ` sψpξqq (deﬁnition of ψ) “ exp p´i ⟨ξ, Xpsq⟩ ` sψpξqq .. (5.191). martingale theory it follows that there exists a cadlag version M ξ “ ˘ `From M ξ ptq : t ě 0 of the martingale t ÞÑ exp p´i ⟨ξ, Xptq⟩ ` tψpξqq. By this we mean that for every pξ, tq P Rν ˆ r0, 8q there exists an event Nt,ξ with the c the equality M ξ ptqpωq “ following properties: P pNt,ξ q “ 0 and for ω R Nt,ξ. 396 Download free eBooks at bookboon.com.

(161) Advanced stochastic processes: Part II 6. STOCHASTIC PROCESSES. Some related results 397. exp p´i ⟨ξ, Xpt, ωq⟩ ` tψpξqq holds. Hence, for every ξ P Rν there exists a P-null set Nξ such that for every t P r0, 8q X Q the equality M ξ ptqpωq “ exp p´i ⟨ξ, Xpt, ωq⟩ ` tψpξqq. holds for all ω R Nξ . In other words for all ξ P Rν the equality: ␣ ( P tω : pξ, ωq P Λu ě P Nξc “ 1 holds. From (5.192) it then follows that ż ż ż ż dξ dP1Λc “ dP 0“ Rν. Ω. Ω. Rν. 1Λc dξ.. (5.192). (5.193). The equality in (5.193) implies that for P b λ-almost all pω, ξq the function t ÞÑ exp p´i ⟨ξ, Xpt, ωq⟩q belongs to D. By Lemma 5.122 we see that P-almost surely the following limits exist for all t ě 0: lim Xpsq and lim Xpsq. sÓt sPQ. Deﬁne the process Y by Y ptq “. sÒt sPQ. lim Xpsq. Then the process Y is cadlag:. sÓt, sPQ. see (the proof of) Lemma 5.103 (b). Furthermore, Xptq “ lim Xpsq (in PsÓt,sPQ. distributional sense), and thus Xptq “ Y ptq P-almost surely. The proof of Theorem 5.123 is now complete. 5.124. Theorem (Dynkin-Hunt). Let pXptq, Ft qtě0 be a cadlag L´evy process with a right-continuous filtration pFt : t ě 0q. Let T : Ω Ñ r0, 8q be a stopping time which is not identically 8. So that P tT ă 8u ą 0. On the event tT ă 8u the process Y “ tY ptq : t ě 0u is defined by Y ptq “ Xpt ` T q ´ XpT q. r the process Y has the same distribution as the process X under (a) Under P P. (b) The σ-fields FT and σ tY psq : s ě 0u are P-independent. Proof. (a) For n P N we write Tn “ 2´n r2n T s. Then pTn : n P Nq is a sequence of stopping times with the following properties: (i) tTn ă 8u “ tT ă 8u; (ii) T ď Tn`1 ď Tn ď T ` 2´n , n P N. Deﬁne the sequence of processes pY n : n P Nq via the formula:. Y n ptq “ Xpt ` Tn q ´ XpTn q op de event tTn ă 8u “ tT ă 8u .. Let now f : pRν qm Ñ C be a bounded continuous function, let A be an event in FT Ď FTn , and let s1 ă ¨ ¨ ¨ ă sm be an increasing sequence of ﬁxed times. Then the following equalities hold: “ ‰ E f pY n ps1 q, Y n ps2 q ´ Y n ps1 q, . . . , Y n psm q ´ Y n psm´1 qq 1AXtT ă8u 8 ÿ “ ` ` ˘ ` ˘ “ E 1AXtTn “k2´n u f X s1 ` k2´n ´ X k2´n , . . . k“0. ` ˘ ` ˘˘‰ , X sm ` k2´m ´ X sm´1 ` k2´m. 397 Download free eBooks at bookboon.com.

(162) Advanced stochastic processes: Part II 398 5. SOME RELATED RESULTS. Some related results. (Xpsj ` k2´n q ´ Xpsj´1 ` k2´n q does not depend on Fsj´1 `k2´n ) “. 8 ÿ. k“0. “ ␣ (‰ P A X Tn “ k2´n. “ ` ` ˘ ` ˘ ` ˘ ` ˘˘‰ E f X s1 ` k2´n ´ X k2´n , . . . , X sm ` k2´m ´ X sm´1 ` k2´m. (Xpsj ` k2´n q ´ Xpsj´1 ` k2´n q has the same distribution as Xpsj q ´ Xpsj´1 q) “. 8 ÿ. k“0. “ ␣ (‰ P A X Tn “ k2´n E rf pX ps1 q ´ X p0q , . . . , X psm q ´ X psm´1 qqs. “ P rA X tTn ă 8us E rf pX ps1 q ´ X p0q , . . . , X psm q ´ X psm´1 qqs. “ P rA X tT ă 8us E rf pX ps1 q ´ X p0q , . . . , X psm q ´ X psm´1 qqs .. (5.194). In (5.194) we let n tend to 8. Since the process X is right-continuous it follows that lim Y n ptq “ Y ptq P-almost surely on the event tT ă 8u. By the continuity nÑ8 of the function f the equality “ ‰ E f pY ps1 q, Y ps2 q ´ Y ps1 q, . . . , Y psm q ´ Y psm´1 qq 1AXtT ă8u “ P rA X tT ă 8us E rf pX ps1 q ´ X p0q , . . . , X psm q ´ X psm´1 qqs. (5.195). follows. By taking, in (5.195), the function f of the form f “ f0 ˝ Vm , where Vm : pRν qm Ñ pRν qm is give by Vm px1 , . . . , xm qq “ px1 , . . . , x1 ` ¨ ¨ ¨ ` xm q we see “ ‰ r f0 pY ps1 q, . . . , Y psm qq 1AXtT ă8u E “ P rA X tT ă 8us E rf0 pXps1 q, . . . , Xpsm qqs .. (5.196). Here f0 : pRν qm Ñ C is an arbitrary bounded continuous function. By passing to limits (5.196) follows for arbitrary bounded Borel measurable functions f0 : pRν qm Ñ C. Via the monotone class theorem the assertion in (a) follows. (b) By taking the function f of the form f “ f0 ˝Vm , where Vm : pRν qm Ñ pRν qm is given by Vm px1 , . . . , xm qq “ px1 , . . . , x1 ` ¨ ¨ ¨ xm q in (5.195), we get “ ‰ r f0 pY ps1 q, . . . , Y psm qq 1AXtT ă8u E “ P rA X tT ă 8us E rf0 pXps1 q, . . . , Xpsm qqs ,. (5.197). r rf0 pY ps1 q, . . . , Y psm qqs “ E rf0 pXps1 q, . . . , Xpsm qqs . E. (5.198). where f0 : pRν qm Ñ C is an arbitrary bounded continuous function. Then choose A “ Ω and divide by P tT ă 8u. We get Inserting the result in (5.198) into (5.197) entails. r rf0 pY ps1 q, . . . , Y psm qq 1A q “ E r rf0 pY ps1 q, . . . , Y psm qqs PpAq. r E. (5.199). From (5.199) it follows that the σ-ﬁeld FT is independent of the one generated by tY psq : s ě 0u: for this employ the monotone class theorem. This completes the proof of Theorem 5.124.. 398 Download free eBooks at bookboon.com. .

(163) Advanced stochastic processes: Part7.IIMARKOV PROCESSES. Some related results 399. 7. Markov processes Let pΩ, F, Pq be a probability space and let X, Y and Z be stochastic variables on Ω with values in a topological Hausdorﬀ space E. We assume that E is locally compact and that E is second countable, or, what is the same, that E satisﬁes the second countability axiom. In other words E has a countable basis for its topology. The space E is supplied with the Borel σ-ﬁeld E and we suppose that the variables X, Y and Z are measurable for the σ-ﬁelds F and E. The symbol PX stands for the image measure on E of the probability P under the mapping X. So PX pBq “ P pX P Bq, B P E. The symbol P ˇˇ is a Y X. probability kernel from Ω to E with the property that ż P ˇˇ px, CqPX pdxq “ P pY P C, X P Bq B. Y X. for all B and C in E. As function of the ﬁrst variable the probability kernel P ˇˇ is PX -almost surely determined. Putting it diﬀerently, the funcY X tion x ÞÑ P ˇˇ px, Cq is the Radon-Nikodym derivative of the measure B ÞÑ Y X. P pY P C, X P Bq with respect to the measure B ÞÑ PX pBq “ P pX P Bq. In the following proposition we collect some useful formulas for (conditional) probability kernels.. Excellent Economics and Business programmes at:. “The perfect start of a successful, international career.” CLICK HERE. to discover why both socially and academically the University of Groningen is one of the best places for a student to be. www.rug.nl/feb/education. 399 Download free eBooks at bookboon.com. Click on the ad to read more.

(164) Advanced stochastic processes: Part II 400 5. SOME RELATED RESULTS. Some related results. 5.125. Proposition. Let pΩ, F, Pq, E, X, Y and Z be as described above. Let g : E ˆ E Ñ C be a bounded measurable function and let B and C belong to E. Then the following equalities hold: ż ż (5.200) gpx, yqP ˇˇ px, dyqPX pdxq “ E pgpX, Y qq ; Y X ż P ˇˇ px, CqPX pdxq “ E p1C pY q, X P Bq ; (5.201) Y X B ˇ ` ˘ (5.202) P ˇˇ pX, Cq “ E 1C pY q ˇ σpXq ; Y X ż P ˇˇ py, CqP ˇˇ px, dyq “ P ˇˇ px, Cq, PX -almost surely, (5.203) Z Y Y X Z X ` ˇ ˘ ` ˇ ˘ provided that E Z ˇ σpY q “ E Z ˇ σpX, Y q .. Proof. The equality in (5.201) follows in fact from the deﬁnition of P ˇˇ . Y X By choosing the function g of the form gpx, yq “ 1B pxq1C pyq in (5.200) we see that (5.200) coincides with (5.201). An arbitrary bounded measurable function g can be approximated by linear combinations of functions of the form px, yq ÞÑ 1B pxq1C pyqi, with B and C in E. Let g : E Ñ E be a bounded measurable function. Then the following equalities hold: ˆ ˙ ż E gpXqP ˇˇ pX, Cq “ gpxqP ˇˇ px, CqPX pdxq “ E pgpXq1C pY qq . (5.204) Y X. Y X. From (5.204) the equality in (5.202) follows. Let g : E Ñ C be a bounded measurable function. Then by, among others, (5.202) the following equalities are true: ż ż gpxq P ˇˇ py, CqP ˇˇ px, dyqPX pdxq Z Y Y X ˙ ˆ ˇ ` ` ˘ ˘ ˇ “ E P ˇ pY, CqgpXq “ E E 1C pZq ˇ σpY q gpXq Z Y ˇ ˇ ˘ ˘ ` ` ˘˘ ` ` “ E E 1C pZq ˇ σpX, Y q gpXq “ E E 1C pZqgpXq ˇ σpX, Y q ż (5.205) “ E p1C pZqgpXqq “ gpxqP ˇˇ px, CqPX pdxq. Z X. From (5.205) the equality in (5.203) follows, and completes the proof of Proposition 5.125. . 5.126. Theorem. Let pΩ, F, Pq and pE, Eq be as above. Let X “ tXptq : t ě 0u be a stochastic process with values in the state space E adapted to the filtration pFt : t ě 0q. So every state variable Xptq is a mapping from Ω to E, measurable for the σ-fields Ft and E. In addition, suppose that the family of operators tϑt : t ě 0u from Ω to Ω satisfies the translation property Xpsq˝ϑt “ Xps`tq for all s and t ě 0. Then the following assertions are equivalent (for the implication (iii) ñ (i) it is assumed that Ft “ σ tXpuq : 0 ď u ď tu): (i) For every C P E and every s and t ě 0 the following equality holds: ˇ ˇ ‰ ‰ “ E 1C pXps ` tqq ˇ Ft “ E 1C pXps ` tqq ˇ σ pXptqq P-almost surely; (5.206) “. 400 Download free eBooks at bookboon.com.

(165) Advanced stochastic processes: Part 7. IIMARKOV PROCESSES. Some related results 401. (ii) For every bounded random variable Y : Ω Ñ C, that is measurable for F8 and E, and for every t ě 0 the following equality holds: ˇ ‰ ˇ ‰ “ “ (5.207) E Y ˝ ϑt ˇ Ft “ E Y ˝ ϑt ˇ σ pXptqq P-almost surely;. (iii) For every m P N and for all pm ` 1q-tuple of bounded Borel measurable functions f0 , . . . , fm : E Ñ C the equality: E rf0 pXp0qq f1 pXps1 qq . . . fm pXpsm qqs ż ż ż “ . . . f0 px0 q f1 px1 q . . . fm pxm q loooomoooon. (5.208). m`1 times. P. ˇ. Xpsm qˇXpsm´1 q. pxm´1 , dxm q . . . P. ˇ. Xps1 qˇXp0q. px0 , dx1 qPXp0q pdx0 q,. holds for every s1 ă ¨ ¨ ¨ ă sm in r0, 8q. If the process X is right-continuous, then (i) and (ii) are also equivalent with the following assertions: (iv) For every bounded Borel measurable function f : E Ñ C and for every stopping time T : Ω Ñ r0, 8s the following equality holds P-almost surely on the event tT ă 8u: ˇ ‰ ˇ “ “ ‰ E f pXps ` T qq ˇ FT “ E f pXps ` T qq ˇ σ pT, XpT qq ;. (v) For every bounded random variable Y : Ω Ñ C, which is measurable for F8 , and for every stopping time T : Ω Ñ r0, 8s the equality ˇ ‰ ˇ “ ‰ “ E Y ˝ ϑT ˇ FT “ E Y ˝ ϑT ˇ σ pT, XpT qq (5.209) holds P-almost surely on the event tT ă 8u.. If the process X is right-continuous and if as filtration the internal history is chosen, then all assertions (i) through (v) are equivalent. Proof. (i) ñ (ii). Upon invoking the monotoneś class theorem it suﬃces to prove (ii) for functions Y : Ω Ñ C of the form Y “ m j“1 fj pXpsj qq, where the functions fj , 1 ď j ď m are bounded and measurable. For m “ 1 (i) is clearly ś equivalent with (ii). Next we prove (ii) for Y “ m`1 j“1 fj pXpsj qq starting from śk (ii), but with Y “ j“1 fj pXpsj qq, with 1 ď k ď m. The equalities below then ś show that (5.207) follows for Y “ m`1 j“1 fj pXpsj qq: « ﬀ m`1 ź ˇ fj pXpsj ` tqq ˇ Ft E j“1. « ˜. “E E “E. « m ź j“1. m`1 ź j“1. ˇ fj pXpsj ` tqq ˇ Fsm `t. ¸. ˇ ˇ Ft. ﬀ. ˇ ` ˘ˇ fj pXpsj ` tqq E fm`1 pXpsm`1 ` tqq ˇ Fsm `t ˇ Ft 401 Download free eBooks at bookboon.com. ﬀ.

(166) Advanced stochastic processes: Part II 402. Some related results. 5. SOME RELATED RESULTS. (the equality in (5.207) for Y “ fm`1 pXpsm`1 qq) « ﬀ m ź ˇ “ ‰ˇ fj pXpsj ` tqq E fm`1 pXpsm`1 ` tqq ˇ σ pX psm ` tqq ˇ Ft “E j“1. ś (the equality in (5.207) for Y “ m j“1 gj pXpsj qq, where gj “ fj , 1 ď j ď m ´ 1, ş ˇ px, dyq) and where gm pxq “ fm pxq fm`1 pyqP ˇ “E. “E. « m ź. «. j“1. m ź j“1. ﬀ ˇ ` ˘ˇ fj pXpsj ` tqq E fm`1 pXpsm`1 ` tqq ˇ σpX psm ` tq ˇ σpXptqq ﬀ ˇ ` ˘ˇ fj pXpsj ` tqq E fm`1 pXpsm`1 ` tqq ˇ Fsm `t ˇ σpXptqq. « ˜. “E E “E. Xpsm`1 `tq Xpsm `tq. «. m ź j“1. m ź j“1. ˇ fj pXpsj ` tqq fm`1 pXpsm`1 ` tqq ˇ Fsm `t. ¸. ﬀ ˇ fj pXpsj ` tqq fm`1 pXpsm`1 ` tqq ˇ σpXptqq .. ﬀ ˇ ˇ σpXptqq. Then observe that (5.210) is the same as (5.207), but for Y “ This proves the implication (i) ñ (ii).. (ii) ñ (i). This implication follows by putting Y “ 1C pXpsqq.. śm`1 j“1. (5.210). fj pXpsj qq.. In the past four years we have drilled. 89,000 km That’s more than twice around the world.. Who are we?. We are the world’s largest oilfield services company1. Working globally—often in remote and challenging locations— we invent, design, engineer, and apply technology to help our customers find and produce oil and gas safely.. Who are we looking for?. Every year, we need thousands of graduates to begin dynamic careers in the following domains: n Engineering, Research and Operations n Geoscience and Petrotechnical n Commercial and Business. What will you be?. careers.slb.com Based on Fortune 500 ranking 2011. Copyright © 2015 Schlumberger. All rights reserved.. 1. 402 Download free eBooks at bookboon.com. Click on the ad to read more.

(167) Advanced stochastic processes: Part 7. IIMARKOV PROCESSES. Some related results 403. (ii) ñ (iii). The equality in (5.208) is correct for m “ 0 and for m “ 1. This is a consequence of Proposition 5.125. Again we will apply induction with respect to m. We assume that (5.208) is correct for m and for the increasing m-tuple s1 ă s2 ă ¨ ¨ ¨ sm . Then we see ﬀ « ˜ « ¸ﬀ m`1 m`1 ź ź ˇ E fj pXpsj qq “ E E fj pXpsj qq ˇ Fsm j“0. “E. “E. «. «. j“0. m ź j“0. m ź j“0. ﬀ. ˇ ˘ fj pXpsj qq E fm`1 pXpsm`1 qq ˇ Fsm `. ˇ ` ` ˘˘ fj pXpsj qq E fm`1 pXpsm`1 qq ˇ σ Xpsm q. (Proposition 5.125) « ż m ź “E fj pXpsj qq fm`1 pxm`1 q P “. ż ż. j“0. ˇ. Xpsm`1 qˇXpsm q. ż. ﬀ ﬀ. pXpsm q, dxm`1 q. ˇ . . . f0 px0 q . . . fm`1 pxm`1 qP pxm , dxm`1 q . . . Xpsm`1 qˇXpsm q loooomoooon m`2 times. P. ˇ. Xps1 qˇXp0q. px0 , dx1 q PXp0q pdx0 q .. (5.211). From the equality in (5.208) for m the equality in (5.200) follows for m ` 1 instead of m. (iii) ñ (i). Let C P E and let s and t ą 0. Starting from (iii) we will prove that the following equality holds P-almost surely: ˇ ˇ ˘ ˘ ` ` (5.212) E 1C pXps ` tqq ˇ Ft “ E 1C pXps ` tqq ˇ σ pXptqq .. Choose 0 ă t1 ă ¨ ¨ ¨ ă tm “ t and choose bounded Borel measurable functions f0 , . . . , fm . The following equality is a consequence of (iii): E pf0 pX0 q . . . fm pXptm qq 1C pXps ` tqqq ż ż ż ż f0 px0 q . . . fm pxm q1C pxm`1 q “ ... looooomooooon m`2 times. ˇ. P. Xps`tqˇXptm q. pxm , dxm`1 q P. ˇ. Xptm qˇXptm´1 q. ˇ px0 , dx1 q PXp0q pdx0 q P Xpt1 qˇXp0q ż ż ż ż “ ... f0 px0 q . . . fm pxm q looooomooooon. pxm´1 , dxm q . . .. m`1 times. P. ˇ. Xps`tqˇXptm q. P. ˇ. pxm , Cq P. Xpt1 qˇXp0q. ˇ. Xptm qˇXptm´1 q. pxm´1 , dxm q . . .. px0 , dx1 q PXp0q pdx0 q. 403 Download free eBooks at bookboon.com.

(168) Advanced stochastic processes: Part II 404. Some related results. 5. SOME RELATED RESULTS. „ “ E f0 pX0 q . . . fm pXptm qq P. ȷ ˇ pXptm q, Cq . Xps`tqˇXptq. (5.213). The monotone class theorem applies to the eﬀect that (5.212) follows from (5.213), provided that the internal history is chosen as ﬁltration. (iv) ñ (v). By the monotoneśclass theorem it suﬃces to prove (ii) for functions Y : Ω Ñ C of the form Y “ m j“1 fj pXpsj qq, where the functions fj , 1 ď j ď m are bounded and measurable. For m “ 1 it is clear that (iv) is equivalent śm`1 to (v). We prove (v) for Y “ j“1 fj pXpsj qq starting from (iv), but with ś Y “ kj“1 fj pXpsj qq, for 1 ď k ď m. The following equalities show that the ś equality (5.209) then follows for Y “ m`1 j“1 fj pXpsj qq: « ﬀ m`1 ź ˇ E fj pXpsj ` T qq ˇ FT j“1. « ˜. “E E. “E. « m ź. m`1 ź j“1. ˇ fj pXpsj ` T qq ˇ Fsm `T. ¸. ˇ ˇ FT. ﬀ. ˇ ‰ˇ fj pXpsj ` T qq E fm`1 pXpsm`1 ` T qq ˇ Fsm `T ˇ FT. j“1. “. ﬀ. (apply equality (5.209) for Y “ fm`1 pXpsm`1 qq) « ﬀ m ź ˇ “ ‰ˇ fj pXpsj ` T qq E fm`1 pXpsm`1 ` T qq ˇ σ psm ` T, X psm ` T qq ˇ FT “E j“1. ś (use equality (5.209) for Y “ m j“1 gj pXpsj qq, where gj “ fj , 1 ď j ď m ´ 1, ş ˇ px, dyq) and where gm pxq “ fm pxq fm`1 pyqP psm `T,Xpsm`1 `T qqˇpsm `T,Xpsm `T qq “E. “E. « m ź j“1. « m ź j“1. “E. « m ź j“1. ﬀ. σpT, XpT qq. « «. “E E. ˇ “ ‰ˇ fj pXpsj ` T qq E fm`1 pXpsm`1 ` T qq ˇ σ pT, X psm ` T qq ˇ. ﬀ ˇ ‰ˇ fj pXpsj ` T qq E fm`1 pXpsm`1 ` T qq ˇ Fsm `T ˇ σpT, XpT qq m ź j“1. “. ˇ fj pXpsj ` T qq fm`1 pXpsm`1 ` T qq ˇ Fsm `T. ﬀ. ﬀ ˇ ˇ σpT, XpT qq. ﬀ ˇ fj pXpsj ` T qq fm`1 pXpsm`1 ` T qq ˇ σpT, XpT qq .. (5.214). ś Then realize that (5.214) is the same as (5.209) for Y “ m`1 j“1 fj pXpsj qq. This proves the implication (iv) ñ (v). The implication (v) ñ (iv) is again trivial.. 404 Download free eBooks at bookboon.com.

(169) Advanced stochastic processes: Part II. 7. MARKOV PROCESSES. Some related results 405. (i) ñ (iv). By the fact E satisﬁes the second countability axiom, and by the fact E is the Borel ﬁeld it suﬃces to prove (iv) for functions f P C0 pEq instead of 1C (verify this precisely). So we have to show the following equality: ˇ ˇ ‰ “ ‰ “ E f pXps ` T qq ˇ FT 1tT ă8u “ E f pXps ` T qq ˇ σpT, XpT qq 1tT ă8u , (5.215). for f P C0 pEq and for s ě 0. By employing the right-continuity of paths, it suﬃces to prove (5.215) for the stopping times Tn :“ 2´n r2n T s, n P N, instead of T . The equality for T then follows from those of Tn by letting n tend to 8. For this notice that 0 ď T ´ Tn`1 ď T ´ Tn ď 2´n . Choose the event A P FTn . Then the event A X tTn “ k2´n u belongs to Fk2´n and the following equalities hold: ␣ “ (‰ E f pXps ` Tn qq , A X Tn “ k2´n ˇ ˘ ␣ (‰ “ ` “ E E f pXps ` Tn qq ˇ Fk2´n , A X Tn “ k2´n ȷ „ż ˘ ␣ ` ( ´n ´n ˇ “E f pyqP Xpk2 q, dy , A X Tn “ k2 Xps`k2´n qˇXpk2´n q „ ȷ ż ˇ pXpTn qpωq, dyq 1tAXtTn “k2´n uu pωq . “ E ω ÞÑ f pyqP ˇ Xps`Tn pωqq XpTn pωqq. (5.216). American online LIGS University is currently enrolling in the Interactive Online BBA, MBA, MSc, DBA and PhD programs:. ▶▶ enroll by September 30th, 2014 and ▶▶ save up to 16% on the tuition! ▶▶ pay in 10 installments / 2 years ▶▶ Interactive Online education ▶▶ visit www.ligsuniversity.com to find out more!. Note: LIGS University is not accredited by any nationally recognized accrediting agency listed by the US Secretary of Education. More info here.. 405 Download free eBooks at bookboon.com. Click on the ad to read more.

(170) Advanced stochastic processes: Part II 406 5. SOME RELATED RESULTS. Some related results. We also have “ ␣ (‰ E f pXps ` Tn qq , A X Tn “ k2´n ˘ˇ ˘ ␣ (‰ “ ` ` “ E E f Xps ` k2´n q ˇ Fk2´n , A X Tn “ k2´n. (because of (i)) ˘ˇ ` ` ˘˘˘ ␣ (‰ “ ` ` , A X Tn “ k2´n “ E E f Xps ` k2´n q ˇ σ X k2´n ȷ „ż ˘ ␣ ` ( ´n ´n ˇ “E Xpk2 q, dy , A X Tn “ k2 f pyqP Xps`k2´n qˇXpk2´n q „ ȷ ż ˇ “ E ω ÞÑ f pyqP pXpTn qpωq, dyq 1tAXtTn “k2´n uu pωq . ˇ Xps`Tn pωqq XpTn pωqq. (5.217). We see that (5.216) and (5.217) are the same. It follows that the assertion in (iv) is proved for Tn instead of T . By letting n tend to 8 we then obtain (iv) for T (by employing the right-continuity of paths of the process). So the proof of Theorem 5.126 is complete now.. . We continue with some deﬁnitions. 5.127. Definition. Let pΩ, F, Pq be a probability space, and let E be a locally compact Hausdorﬀ space with a countable basis for its topology. In addition, let pFt : t ě 0q be a ﬁltration on Ω. Let X “ tXptq : t ě 0u be a process attaining values in E. The state space E is equipped with the Borel ﬁled and it is assumed that X is an adapted process. Suppose that for every x P E the (sub-)probability ˇ px, Cq, C P E, is deﬁned. Here the (sub-)probability kernel kernel P Xps`tqˇXptq PY |X px, Cq possesses the following deﬁning property: ż P ˇˇ px, CqP pX P dxq “ P tY P C, X P Bu , B. Y X. where B and C are Borel subsets of E and where X and Y are stochastic variables with values in E. In addition, it is assumed that there are so-called translation operators ϑt : Ω Ñ Ω with the property that Xpsq˝ϑt “ Xps`tq for all s, t ě 0. Moreover, by hypothesis the process X is cadlag. We say that the process X is a Markov process if for every C P E and every t ě 0 the equality ˇ ‰ ˇ “ ‰ “ E 1C pXps ` tqq ˇ Ft “ E 1C pXps ` tqq ˇ σpXptqq (5.218). is P-almost surely true for all s ě 0. The process X is called a strong Markov process if equality (5.218) also holds for stopping times. More precisely, if for every s ě 0, for every C P E and for every stopping time T : Ω Ñ r0, 8s the equality ˇ ‰ ˇ “ ‰ “ E 1C pXps ` T qq ˇ FT “ E 1C pXps ` T qq ˇ σpT, XpT qq. holds P-almost surely on the event tT ă 8u. If the process X is cadlag is, then a Markov process is automatically a strong Markov: see Theorem 5.126. We. 406 Download free eBooks at bookboon.com.

(171) Advanced stochastic processes: Part 7. IIMARKOV PROCESSES. Some related results 407. say that a Markov process X is time homogeneous if for all C P E and for all s and t ě 0 the equality ˇ px, Cq “ P ˇ P px, Cq (5.219) Xps`tqˇXptq XpsqˇXp0q. is true for all x P E. In what follows we always suppose that X is a cadlag, time homogeneous Markov process. Furthermore we define the operators tP ptq : t ě 0u via the formula ż ˇ rP psqf s pxq “ f pyqP px, dyq. (5.220) ˇ Xpsq Xp0q. Here s ě 0 and f belongs to C0 pEq. Since we have (see equality (5.203) in Proposition 5.125) ż ˇ py, CqP ˇ ˇ P px, dyq “ P px, Cq, PXp0q -almost surely, ˇ ˇ ˇ Xps`tq Xptq. Xptq Xp0q. Xps`tq Xp0q. (5.221) we get, for a time-homogeneous Markov process X the following equalities: ż ˇ px, dyq rP psqP ptqf s pxq “ rP ptqf s pyqP XpsqˇXp0q ż ż ˇ “ f pzqP ˇˇ py, dzqP px, dyq ˇ Xptq Xp0q. Xpsq Xp0q. (X is time homogeneous). “. ż ż. f pzqP. ˇ. py, dzqP. ż ż. f pzqP. ˇ. px, dzq “ rP ps ` tqf s pxq. (employ equality (5.221)) “. Xps`tqˇXpsq Xps`tqˇXp0q. ˇ. XpsqˇXp0q. px, dyq. The cadlag property of X implies limsÓ0 rP psqf s pxq “ f pxq for all f P C0 pEq and for all x P E. If P psqf belongs to C0 pEq for every f P C0 pEq and for every s ě 0, then the family tP ptq : t ě 0u apparently constitutes a Feller semigroup. Put ˇ px, Cq, s ě 0, x P E, C P E. Let the expectation values P ps, x, Cq “ P XpsqˇXp0q ś of Ex pY q, x P E, Y “ m j“1 fj pX psj qq, s1 ă s2 ă . . . ă sm , be determined by the formula: ¸ ˜ m ź fj pXpsj qq Ex “. ż. j“1. .... ż ź m j“1. fj pxj q P ps1 , x, dx1 q . . . P psm ´ sm´1 , xm´1 , dxm q .. (5.222). “ ˇ ‰ Instead of (5.222) most of the time we write Ex pY q “ E Y ˇ Xp0q “ x , for a bounded stochastic variable Y . Since X is a time homogeneous Markov process we see that the following equality also holds Px -almost surely: ˇ ˘ ` Ex Y ˝ ϑt ˇ Ft “ EXptq pY q, (5.223) 407 Download free eBooks at bookboon.com.

(172) Advanced stochastic processes: Part II 408 5. SOME RELATED RESULTS. Some related results. for all t ě 0 and for all bounded random variables Y . The ś equality in (5.223) is ﬁrst proved for random variables Y of the form Y “ m j“1 fj pXpsj qq, where the functions fj , 1 ď j ď m, are bounded Borel functions. Equality (5.223) is also true if Px and Ex are replaced by P and E respectively. 5.128. Remark. The expectation value Ex pY q is in fact the Radon-Nikodym derivative of de measure B ÞÑ E rY, Xp0q P Bs with respect to the measure B ÞÑ P rXp0q P Bs. If in this deﬁnition we take for Y the variable Y “ 1C pXpsqq, ˇ px, Cq. Hence, these quantithen we obtain the probability kernel P ˇ Xpsq Xp0. ties are deﬁned as Radon-Nikodym derivatives. So, in general, the expression ˇ px, Cq is not deﬁned for every x P E. However, we will assume that P XpsqˇXp0q these probability kernels exist for every x P E indeed, and that the corresponding semigroup is a Feller. Many authors deﬁne a (time homogeneous) Markov process X relative to a family of probability measures tPx : x P Eu by means of the following equality: ˇ ˘ ` Ex Y ˝ ϑt ˇ Ft “ EXptq pY q , (5.224) Px -almost surely for all x P E, for all t ě 0 and for all bounded random variables Y : Ω Ñ C. In fact we also do this. In the time homogeneous case the equality in (5.224) also holds for stopping times T : ˇ ˘ ` (5.225) Ex Y ˝ ϑT ˇ FT “ EXpT q pY q , Px -almost surely on the event tT ă 8u, provided that the process X is cadlag.. .. 408 Download free eBooks at bookboon.com. Click on the ad to read more.

(173) Advanced stochastic processes: Part II 8. DOOB-MEYER DECOMPOSITION. Some related results 409. The equality in (5.225) can be proved in the same manner as equality (5.209) in Theorem 5.126. Therefore pick f P C0 pEq and a stopping time T : Ω Ñ r0, 8s. Consider the stopping times Tn :“ 2´n r2n T s, n P N, instead of T . Then, for an event A P FTn , we have “ ‰ Ex f pXps ` Tn qq 1AXtTn “k2´n u ˘ “ ` ‰ “ Ex f Xps ` k2´n q 1AXtTn “k2´n u ˘ˇ “ ` ` ˘ ‰ “ Ex Ex f Xps ` k2´n q ˇ Fk2´n 1AXtTn “k2´n u ˘ˇ “ ` ` ˘ ‰ “ Ex Ex f Xps ` k2´n q ˇ Fk2´n 1AXtTn “k2´n u “ ‰ “ Ex EXpk2´n q pf pXpsqqq 1AXtTn “k2´n u “ ‰ (5.226) “ Ex EXpTn q pf pXpsqqq 1AXtTn “k2´n u . From (5.226) it follows that (5.225) for Y “ f pXpsqq and for Tn in the place of T . By taking the limit in (5.226) for n Ñ 8 the equality in (5.225) follows for Y “ f pXpsqq. Precisely as in the proof of the implication (iv) ñ (v) in Theorem 5.126 the equality in (5.225) then follows for arbitrary random variables Y : Ω Ñ C, which are bounded and measurable for the σ-ﬁeld F8 . 8. The Doob-Meyer decomposition via Komlos theorem Let pΩ, F, Pq be a probability space, let tFt : t ě 0u be a right continuous ﬁltration in F and let tXptq : t ě 0u be a real-valued Ft -submartingale. The DoobMeyer decomposition theorem states that there exists an Ft -martingale tM ptq : t ě 0u together with an increasing predictable adapted process tAptq : t ě 0u, which is right continuous P-almost surely, such that Xptq “ M ptq ` Aptq, t ě 0, provided that the process tXptq : t ě 0u is of class (DL). The latter means that for every t ą 0 the family tXpτ q : 0 ď τ ď t, τ stopping timeu is uniformly integrable. Moreover this decomposition is unique in case we assume that Ap0q “ 0. By Doob’s optional sampling theorem every martingale is automatically of class (DL) (see e.g. Ikeda and Watanabe [61], p.35, Ethier and Kurtz [54], p.74). An interesting discussion of the Doob-Meyer decomposition and (sub-)martingale theory can be found in Kopp [74]. For a nice account of the Doob-Meyer decomposition theorem the reader may also consult van Neerven [148]. We shall employ the following result of Komlos [73]. In fact it can be interpreted as kind of a law of large numbers. 5.129. Theorem (Komlos). Let tfk : k P Nu be a sequence in L1 pΩ, F, Pq such that sup tE p|fk |q : k P Nu ă 8.. Then there exists an infinite large subset Λ0 of N together with a function f in L1 pΩ, F, Pq such that for every infinite subset Λ of Λ0 ÿ 1 lim fj “ f, P-almost surely. (5.227) jPΛXr1,ns nÑ8 |Λ X r1, ns| 409 Download free eBooks at bookboon.com.

(174) Advanced stochastic processes: Part II 410 5. SOME RELATED RESULTS. Some related results. Examples show that this limit need not be an L1 -limit. Set Ω “ N with the discrete σ-ﬁeld and with Ptku “ 2´k , k P N. Let tfk : k P Nu be the sequence ek , k P N, where tek : k P Nuş is deﬁned by fk “ 2kř řnthe sequence of the unit vectors. Then n´1 nj“1 fj Ñ 0 pointwise, but n´1 j“1 fj dP “ 1, n P N.. Standard results on continuity properties of submartingales yield the existence of a realization (version) which is continuous from the right and possesses left limits P-almost surely. Henceforth we shall assume that the Ft -submartingale tXptq : t ě 0u is continuous from the right and has left limits P-almost surely. We shall prove that there exists a predictable increasing process tAptq : t ě 0u together with an inﬁnite Λ0 of N such that for every inﬁnite subset Λ of Λ0 and every t ě 0 the variable Aptq is given as the limit: ÿ 1 Aptq “ lim Aj ptq, (5.228) jPΛXr1,ns nÑ8 |Λ X r1, ns| where. Aj ptq “. ÿ. " ˆ ˆ ˙ ˙ ˆ ˙* k`1 k ´j E X | F ´ X . k2 j 0ďkă2 t 2j 2j. (5.229). Moreover the process tXptq ´ Aptq : t ě 0u is an Ft -martingale. The limit in (5.228) is a point-wise almost sure limit as well as an L1 -limit. Again let pΩ, F, Pq be a probability space, let tFt : t ě 0u be a right-continuous ﬁltration in F and let tXptq : t ě 0u be right continuous submartingale of class (DL) which possesses almost sure left limits. We want to prove the following version of the Doob-Meyer decomposition theorem. 5.130. Theorem. There exists a unique predictable right continuous increasing process tAptq : t ě 0u with Ap0q “ 0 such that the process tXptq ´ Aptq : t ě 0u is an Ft -martingale. It is perhaps useful to insert the following proposition. 5.131. Proposition. Processes of the form M ptq ` Aptq, with M a martingale and with A an increasing process in L1 pΩ, F, Pq are of class (DL). Proof of Proposition 5.131. Let tXptq “ M ptq ` Aptq : t ě 0u be the decomposition of the submartingale tXptq : t ě 0u in a martingale tM ptq : t ě 0u and an increasing process tAptq : t ě 0u with Ap0q “ 0 and 0 ď τ ď t be any Ft -stopping time. Here t is some ﬁxed time. For N P N we have E p|Xpτ q| : |Xpτ q| ě N q ď E p|M pτ q| : |Xpτ q| ě N q ` E pApτ q : |Xpτ q| ě N q ď E p|M ptq| : |Xpτ q| ě N q ` E pApτ q : |Xpτ q| ě N q ď E p|M ptq| ` Aptq : |Xpτ q| ě N q ˙ ˆ ď E |M ptq| ` Aptq : sup |Xpsq| ě N . 0ďsďt. 410 Download free eBooks at bookboon.com. (5.230).

(175) Advanced stochastic processes: Part II. Some related results. 8. DOOB-MEYER DECOMPOSITION. 411. Since N ˆ P tsup0ďsďt |Xpsq| ě N u ď E p|Xptq|q, it follows that lim sup tE p|Xpτ q| : |Xpτ q| ě N q : 0 ď τ ď t, τ. N Ñ8. stopping timeu “ 0. (5.231) . This shows Proposition 5.131 Similarly we have the following result.. 5.132. Proposition. Let tXptq : t ě 0u be an Ft -submartingale. For any real number N the process tmaxpXptq, N q : t ě 0u is an Ft -submartingale which is of class (DL). Next we come to the heart of the matter. The symbol rxs, x P R, denotes the integer k with k ă x ď k ` 1.. Join the best at the Maastricht University School of Business and Economics!. Top master’s programmes • 3 3rd place Financial Times worldwide ranking: MSc International Business • 1st place: MSc International Business • 1st place: MSc Financial Economics • 2nd place: MSc Management of Learning • 2nd place: MSc Economics • 2nd place: MSc Econometrics and Operations Research • 2nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier ‘Beste Studies’ ranking 2012; Financial Times Global Masters in Management ranking 2012. Maastricht University is the best specialist university in the Netherlands (Elsevier). Visit us and find out why we are the best! Master’s Open Day: 22 February 2014. www.mastersopenday.nl. 411 Download free eBooks at bookboon.com. Click on the ad to read more.

(176) Advanced stochastic processes: Part II 412 5. SOME RELATED RESULTS. Some related results. Proof of Theorem 5.130. It will be convenient to introduce the following processes: ˙ ˆ ˆ j ˙ r2 ts | Ft , t ě 0, j P N; (5.232) Xj ptq “ E X 2j " ˆ ˆ ˙ ˙ ˆ ˙* ÿ k`1 k E X | Fk2´j ´ X . (5.233) Aj ptq “ j 0ďkă2j t 2 2j. The processes tAj ptq : t ě 0u are right continuous and have left limits. The processes tAj ptq : t ě 0u are predictable in the sense that, for j, N in N, the functions pt, ωq ÞÑ␣Aj pt, ωq are measurable with (respect to the σ-field generated by the collection 1pa,bs ˆ A : 0 ď a ă b, A P Fa : see e.g. Durrett [44], p. 49. Moreover it is readily verified that the process tXj ptq ´ Aj ptq : t ě 0u. (5.234). is an Ft -martingale and that lim E pAj ptq ´ Aj pt´qq “ 0.. jÑ8. (5.235). Equality (5.235) is true because lim E pXpsqq “ E pXptqq . sÓt. (5.236). Equality (5.236) can be proved in the following manner. Put X 2 ptq “ lim E pXpt ` hq | Ft q “ inf E pXpt ` hq | Ft q . hÓ0. hą0. Then X 2 ptq ě Xptq, P-almost surely. The following argument shows that X 2 ptq “ Xptq, P-almost surely. Define for m P N the stopping time τm by τm “ infts ą 0 : |Xpsq| ą mu.. Then, P-almost surely, τm Ò 8. Moreover, we have E rX 2 ptq ´ Xptq : τm ą ts. (5.237). “ lim E rE pXpt ` hq | Ft q ´ Xptq : τm ą ts hÓ0 “ ` ˘ ‰ “ lim E E pXpt ` hq ´ Xptqq 1tτm ątu | Ft hÓ0. “ lim E rXpt ` hq ´ Xptq : τm ą ts hÓ0. “ lim tE rXpt ` hq ´ Xptq : τm ą t ` hs hÓ0. `E rXpt ` hq ´ Xptq : t ă τm ď t ` hsu. ď lim tE rXpt ` hq ´ Xptq : τm ą t ` hs hÓ0. `E rE pXpt ` 1q | Ft`h q ´ Xptq : t ă τm ď t ` hsu. “ lim tE rXpt ` hq ´ Xptq : τm ą t ` hs hÓ0. `E rXpt ` 1q ´ Xptq : t ă τm ď t ` hsu “ 0,. 412 Download free eBooks at bookboon.com.

(177) Advanced stochastic processes: Part II 8. DOOB-MEYER DECOMPOSITION. Some related results 413. by dominated convergence (twice: on tτm ą t ` hu we have |Xpt ` hq ´ Xptq| ď 2m, P-almost surely). Consequently 0 ď E pX 2 ptq ´ Xptqq “ lim E pX 2 ptq ´ Xptq : τm ą tq “ 0, mÑ8. (5.238). and hence X 2 ptq “ Xptq, P-almost surely. We also infer ˆ ˙ 2 EpXptqq “ EpX ptqq “ E lim E pXpt ` hq | Ft q ´ Xptq ` EpXptqq hÓ0 ˆ ˙ “ E lim E ppXpt ` hq ´ Xptqq | Ft q ` EpXptqq hÓ0. “ lim E pE ppXpt ` hq ´ Xptqq | Ft qq ` EpXptqq hÓ0. “ lim E pXpt ` hq ´ Xptqq ` EpXptqq “ lim E pXpt ` hqq . hÓ0. hÓ0. (5.239). This proves (5.236). In addition we write f ptq “ E pXptq ´ Xp0qq. (5.240). D “ tt ě 0 : t P Qu Y tt ě 0 : f pt`q ą f pt´qu .. (5.241). and we deﬁne the countable dense subset D of r0, 8q by (Notice that the functions f is increasing.). Let Λ0 be any inﬁnite subset of N and let tAΛ0 ptq : t P Du be a process such that for every inﬁnite subset Λ of Λ0 and P-almost surely, ÿ 1 Aj ptq, t P D. (5.242) AΛ0 ptq “ lim jPΛXr1,ns nÑ8 |Λ X r1, ns|. By Komlos’ theorem (Theorem 5.129) and a diagonal procedure such a subset Λ0 exists. We shall prove that for t P D the limit in (5.241) also exists in L1 sense. In view of a theorem of Scheﬀ´e (Corollary 2.12.5 in Bauer [10], p. 105, it suﬃces to prove that ÿ 1 E pAj ptqq . (5.243) E pAΛ0 ptqq “ lim jPΛXr1,ns nÑ8 |Λ X r1, ns| It is readily veriﬁed that. ˆ. E pAj ptqq “ E X so that. ˆ. r2j ts 2j. ˙˙. ´ E pXp0qq ,. ÿ 1 E pAj ptqq “ f pt`q, jPΛXr1,ns nÑ8 |Λ X r1, ns| lim. t P D.q. On the other hand we have, by Fatou’s lemma, ÿ 1 E pAj ptqq “ f pt`q. E pAΛ0 ptqq ď lim inf jPΛXr1,ns nÑ8 |Λ X r1, ns| In addition we have for λ ą 0 ˆ ˙ ÿ 1 E pAΛ0 ptqq ě E lim sup Aj ptq jPΛXr1,ns nÑ8 |Λ X r1, ns|. 413 Download free eBooks at bookboon.com. (5.244). (5.245). (5.246).

(178) Advanced stochastic processes: Part II 414. Some related results. 5. SOME RELATED RESULTS. ˆ. ˙ ÿ 1 ě E lim sup Aj pmin pt, τλ q ´q , (5.247) jPΛXr1,ns nÑ8 |Λ X r1, ns| where τλ is the stopping time defined by " * ÿ 1 τλ “ inf s ą 0 : s P D, sup Aj psq ě λ . (5.248) jPΛXr1,ns nPN |Λ X r1, ns| Since ÿ 1 Aj pmin pt, τλ q ´q ď λ, (5.249) |Λ X r1, ns| jPΛXr1,ns we infer from (5.247) and (5.235) that, for any λ ą 0, ÿ 1 E pAj pmin pt, τλ q ´qq E pAΛ0 ptqq ě lim sup jPΛXr1,ns nÑ8 |Λ X r1, ns| ě E pXptq : τλ ą tq ` E pX pminpτλ , tqq : τλ ď tq ´ E pXp0qq . (5.250) Since τλ Ò 8, P-almost surely, as λ tends to infinity, we infer from (5.250) together with the fact that the collection tXpτ q : τ ď t, τ stopping timeu is uniformly integrable, E pAΛ0 ptqq ě E pXptq ´ Xp0qq “ f ptq.. (5.251). > Apply now redefine your future. - © Photononstop. AxA globAl grAduAte progrAm 2015. axa_ad_grad_prog_170x115.indd 1. 19/12/13 16:36. 414 Download free eBooks at bookboon.com. Click on the ad to read more.

(179) Advanced stochastic processes: Part II 8. DOOB-MEYER DECOMPOSITION. Some related results 415. (In fact in (5.250) we ﬁrst take the sum, then we write Ω “ tτλ ą tu Y tτλ ď tu.) The right continuity of the submartingale tXptq : t ě 0u together with (5.236) implies the equality f ptq “ f pt`q. (5.252) Hence the equality in (5.243) now follows from (5.251), (5.252) and (5.246). So the limit in (5.241) is also an L1 -limit. Since the submartingale tXptq : t ě 0u is continuous from the right we also deduce ˆ ˆ j ˙˙ r2 ts ´ E pXptqq lim sup E p|Xj ptq ´ Xptq|q “ lim sup E X 2j jÑ8 jÑ8 “ lim f psq ´ f ptq “ 0. (5.253) sÓt. Hence the L1 -convergence in the equality ÿ 1 Xj ptq |Λ X r1, ns| jPΛXr1,ns ÿ ÿ 1 1 Mj ptq ` Aj ptq “ |Λ X r1, ns| jPΛXr1,ns |Λ X r1, ns| jPΛXr1,ns yields (5.254) Xptq “ MΛ0 ptq ` AΛ0 ptq, t P D, where the process tAΛ0 ptq : t P Du is increasing and predictable. We shall extend (5.254) to all t ě 0 and we shall prove that the process tAΛ0 ptq : t P Du has right continuous extensions to all of r0, 8q. In order to achieve this ﬁx t0 R D, t0 ą 0, and let s, t be arbitrary numbers in D with 0 ă s ă t0 ă t ă 8. Then ÿ 1 AΛ0 psq ď lim inf Aj psq jPΛXr1,ns nÑ8 |Λ X r1, ns| ÿ 1 ď lim inf Aj pt0 q jPΛXr1,ns nÑ8 |Λ X r1, ns| ÿ 1 Aj ptq ď AΛ0 ptq. (5.255) ď lim sup jPΛXr1,ns nÑ8 |Λ X r1, ns|. From (5.255) it follows that ˙ ˆ ÿ ÿ 1 1 Aj pt0 q ´ lim inf Aj pt0 q E lim sup jPΛXr1,ns jPΛXr1,ns nÑ8 |Λ X r1, ns| nÑ8 |Λ X r1, ns| (5.256) ď E pAΛ0 ptq ´ AΛ0 psqq “ E pXptq ´ Xpsqq “ f ptq ´ f psq. So that ˆ E lim sup. ˙ ÿ ÿ 1 1 Aj pt0 q ´ lim inf Aj pt0 q jPΛXr1,ns jPΛXr1,ns nÑ8 |Λ X r1, ns| nÑ8 |Λ X r1, ns| (5.257) ď f pt0 `q ´ f pt0 ´q “ f pt0 q ´ f pt0 q “ 0,. since t0 does not belong to D. Hence, for every t0 ě 0, ÿ 1 Aj pt0 q AΛ0 pt0 q “ lim sup jPΛXr1,ns nÑ8 |Λ X r1, ns| ÿ 1 Aj pt0 q, “ lim inf jPΛXr1,ns nÑ8 |Λ X r1, ns|. 415 Download free eBooks at bookboon.com. (5.258).

(180) Advanced stochastic processes: Part II 416. 5. SOME RELATED RESULTS. Some related results. P-almost surely. In addition, as above we also have E pAΛ0 ptqq “ f ptq,. t ě 0, q. (5.259). and hence E pAΛ0 ptq ´ AΛ0 psqq “ f ptq ´ f psq.. (5.260). So that the process tAΛ0 ptq : t ě 0u is almost surely right continuous. Again we have decomposition (5.254) for all t ě 0. From (5.236) and (5.258) it follows that, P-almost surely, ÿ 1 AΛ0 pt0 q “ lim Aj pt0 ´q jPΛXr1,ns nÑ8 |Λ X r1, ns|. and consequently the process tAΛ0 ptq : t ě 0u is predictable.. . The uniqueness of the Doob-Meyer decomposition does not depend on the (DL)property. So the processes tMΛ0 ptq : t ě 0u and tAΛ0 ptq : t ě 0u do not depend on the particular choice of Λ0 . Henceforth we write Xptq “ M ptq ` Aptq,. t ě 0,. (5.261). where tM ptq : t ě 0u is an Ft -martingale and where tAptq : t ě 0u is an increasing right continuous process which is predictable. Proposition 5.131 shows that the process tXptq : t ě 0u must possess the (DL)-property. Let D0 be the countable dense subset of r0, 8q given by D0 “ tt P Q : t ě 0u Y tt ě 0 : f pt`q ą f pt´qu. (5.262). and choose Λ0 Ď N, |Λ0 | “ 8, and the process tBptq : t P D0 u in such a way that for every infinite subset Λ of Λ0 , ÿ 1 Bptq “ lim Aj ptq, t P D0 . (5.263) jPΛXr1,ns nÑ8 |Λ X r1, ns|. Then, as in the case of tAj ptq : j P Nu it follows that the convergence in (5.263) is an L1 -convergence as well. Again as above the convergence in (5.263) occurs for all t ě 0. Consequently the process tXptq ´ Bptq : t ě 0u is a martingale, because the processes tXj ptq ´ Aj ptq : t ě 0u, j P N, are martingales. Here ȷ „ ˆ j ˙ r2 ts Xj ptq “ E X | Ft . 2j These remarks prove the following corollary. 5.133. Corollary. Write a submartingale tXptq : t ě 0u in the form Xptq “ M ptq ` Aptq, t ě 0, where the process tM ptq : t ě 0u is a martingale and where tAptq : t ě 0u is a right continuous increasing predictable process with Ap0q “ 0. Then there exists an infinite subset Λ0 of N such that for every infinite subset Λ of Λ0 and every t ě 0: ÿ 1 Aptq “ lim Aj ptq. (5.264) jPΛXr1,ns nÑ8 |Λ X r1, ns| 416 Download free eBooks at bookboon.com.

(181) Advanced stochastic processes: Part II 8. DOOB-MEYER DECOMPOSITION. Here. Some related results 417. ˆ ˆ ˆ ˙ ˙ ˆ ˙˙ k`1 k ´j E X | F ´ X Aj ptq “ k2 0ďkă2j t 2j 2j and the convergence in (5.264) is a P-almost sure as well as an L1 -convergence. Of course the process tAptq : t ě 0u does not depend on the particular choice of Λ0 for which all the limits in (5.264) exist. ÿ. Next the uniqueness part of the Doob-Meyer decomposition will follow from Proposition 5.134. 5.134. Proposition. Let Z “ tZptq : t ě 0u be a bounded martingale and let A “ tAptq : t ě 0u and tBptq : t ě 0u be adapted increasing processes such that B ´ A is a martingale. Also suppose that EpAptqq ă 8, for t ě 0. Then E rZpt`q pBpt`q ´ Apt`qq ´ Zp0q pBp0q ´ Ap0qqs ˙ ˆż t pZps`q ´ Zps´qq dpB ´ Aqpsq . “E. (5.265). 0. şt 5.135. Remark. The integral 0 pZps`q ´ Zps´qq dpB ´ Aqpsq should be interpreted as follows: żt pZps`q ´ Zps´qq dpB ´ Aqpsq 0 ż8 ż8 “ pZps`q ´ Zps´qq 1p0,ts psqdBpsq ´ pZps`q ´ Zps´qq 1p0,ts psqdApsq. 0. 0. Proof of Proposition 5.134. Let n P N. Since Z is a martingale we have: ˘` ` ˘ ` ˘˘‰ “ ` ´ E rZp0q pBp0q ´ Ap0qqs E Z r2n ts2´n B r2n ts2´n ´ A r2n ts2´n » ÿ ` ˘ “ E– Z pj ` 1q2´n 0ďjăr2n ts. ﬁ. ˘ ` ˘˘ ` ` ˘ ` ˘˘( ␣` ` B pj ` 1q2´n ´ B j2´n ´ A pj ` 1q2´n ´ A j2´n ﬂ .. Since B ´ A is a martingale, it follows that: ˘‰ “ ` ˘` ` ˘ ` ˘ E Z r2n ts2´n B r2n ts2´n ´ Bp0q ´ A r2n ts2´n ` Ap0q. Put. ´ E rZp0q pBp0q ´ Ap0qqs ´ÿ ` ` ˘ ` ˘˘ “E Z pj ` 1q2´n ´ Z j2´n n 0ďjăr2 ts `` ` ˘ ` ˘˘ ` ` ˘ ` ˘˘˘¯ B pj ` 1q2´n ´ B j2´n ´ A pj ` 1q2´n ´ A j2´n . ` ` ˘ ÿ8 ˘ Z pj ` 1q2´n 1pj2´n ,pj`1q2´n s psq, and Zn` psq “ Z r2n ss2´n “ j“0 ` n ` ˘ ÿ8 ˘ ´ ´n Zn psq “ Z pr2 ss ´ 1q 2 Z j2´n 1pj2´n ,pj`1q2´n s psq. “ j“0. 417 Download free eBooks at bookboon.com.

(182) Advanced stochastic processes: Part II 418 5. SOME RELATED RESULTS. Some related results. Then we obtain ` ` ˘` ` ˘ ` ˘˘˘ E Z r2n ts2´n B r2n ts2´n ´ A r2n ts2´n ´ E rZp0q pBp0q ´ Ap0qqs ˙ ˆż 8 ˘ ` ` ´ Zn psq ´ Zn psq 1p0,r2n ts2´n s psqd pBpsq ´ Apsqq . “E 0. So, upon letting n tend to inﬁnity, Proposition 5.134 follows.. . 5.136. Proposition. In addition to the hypotheses in Proposition 5.134, suppose that the martingale B ´ A is predictable. Then Bpt`q “ Apt`q P-almost surely. So that, if B ´ A is right-continuous, then B “ A P-almost surely, provided Bp0q “ Ap0q “ 0. ¯. ´ş t. Proof. First we prove that E 0 pZps`q ´ Zps´qq dpB ´ Aqpsq “ 0. Here we shall employ the predictability of the process B ´A. It suﬃces to prove that, for all s ą 0, E ppZps`q ´ Zps´qq pBps`q ´ Aps`q ´ Bps´q ` Aps´qqq “ 0.. (5.266). Since the predictable ﬁeld on Ωˆr0, 8q is generated by the collection tC ˆpa, bs : C P Fa , 0 ď a ă bu it suﬃces to prove (5.266) for all s ě 0 if B ´ A is of the form Bpsq ´ Apsq “ 1C ˆ 1pa`ε,8q psq, C P Fa . (5.267) So let C belong to Fa and let Bpsq ´ Apsq “ 1C ˆ 1pa`ε,8q psq. Then, for s “ a ` ε (and C P Fa ), we have by the martingale property of Z, E pZ ppa ` εq`q ´ Z ppa ` εq´q 1C q. “ E pE pZpppa ` εq`q ´ Z ppa ` εq´q | Fa q 1C q “ E ppZpaq ´ Zpaqq 1C q “ 0.. (5.268). Notice that, for s “ a ` ε,. E ppZps`q ´ Zps´qq pBps`q ´ Aps`q ´ Bpsq ` Apsqqq “ E ppZ ppa ` εq`q ´ Z ppa ` εq´qq 1C q .. From Proposition 5.134 it now follows that. E pZpt`q pBpt`q ´ Apt`qqq “ E rZp0q pBp0q ´ Ap0qqs “ 0.. Next, ﬁx t ą 0 and deﬁne the martingale Zpsq by ˙ ˆ Bpt`q ´ Apt`q | Fs . Zpsq “ E |Bpt`q ´ Apt`q| ` 1 Then ˜ 0 “ E pZpt`q pBpt`q ´ Apt`qqq “ E. 2. |Bpt`q ´ Apt`q| |Bpt`q ´ Apt`q| ` 1. ¸. and hence Bpt`q “ Apt`q, P-almost surely for all t ě 0. It also follows that Bpt´q “ Apt´q, P-almost surely for all t ą 0. If the process B ´ A is right continuous almost surely, we infer Bptq “ Aptq, t ě 0, P-almost surely. This completes the proof of Proposition 5.136. . 418 Download free eBooks at bookboon.com.

(183) Advanced stochastic processes: Part II 8. DOOB-MEYER DECOMPOSITION. Some related results 419. As a special case the following result contains the uniqueness part of the DoobMeyer decomposition theorem. 5.137. Proposition. Let A and B be increasing adapted processes. Suppose that B ´ A is a predictable right continuous martingale. Then Bptq “ Aptq ` Bp0q ´ Ap0q, P-almost surely. Proof. This result is an immediate consequence of Proposition 5.134 and Proposition 5.136. 5.138. Corollary. There is only one way to write a semi-martingale Y in the form Y “ M ` A, where M is a (local) martingale and where A is a predictable right continuous process of finite variation locally with Ap0q “ 0. 5.139. Remark. An increasing, predictable right continuous real-valued process tAptq : t ě 0u, with E pAptqq ă 8 for t ě 0, is called a Meyer process. It is perhaps worthwhile to isolate the following result in the existence part of Doob-Meyer decomposition theorem: notation is that of the proof of Theorem 1 ÿ An ptq, for a finite subset Λ of N. 5.130. We also use AΛ ptq “ |Λ| nPΛ. 419 Download free eBooks at bookboon.com. Click on the ad to read more.

(184) Advanced stochastic processes: Part II 420 5. SOME RELATED RESULTS. Some related results. 5.140. Theorem. Let tXptq : t ě 0u be a right continuous submartingale of class (DL). For every infinite subset Λ0 of N there exists an infinite subset Λ of Λ0 such that for every further infinite subset Λ1 of Λ, the limit AΛ ptq :“ lim AΛ1 Xr1,N s ptq, N Ñ8. exists P-almost surely,. 1 and does not depend on the choice we are dealing with ˇ‰ “ˇ of Λ . Moreover, since ˇ ˇ 1 (DL)-submartingales, limN Ñ8 E AΛ ptq ´ AΛ Xr1,N s ptq “ 0. In addition, the process tAΛ ptq : t ě 0u is predictable and right continuous.. Proof. Write Q1 “ ttℓ : ℓ P Nu “ tt ě 0 : E pXpt`qq ą E pXpt´qqu. Deﬁne the measure µ on Q1 by ÿ 1 1 µ pIq “ . ℓ 2 1 ` E pXptℓ q ´ Xp0qq ℓPI. ď. pQ X r0, 8qq .. Let Λ0 be an inﬁnite subset of N. Komlos’ theorem, applied to the sequence tAn ptℓ q : ℓ P NunPN on the measure space tN ˆ Ω, PpNq b F, µ b Pu applies to the eﬀect that there exists an inﬁnite subset Λ of Λ0 such that for every further inﬁnite subset Λ1 of Λ, AΛ ptq “ limN Ñ8 AΛ1 Xr1,N s ptℓ q exists for ℓ “ 1, 2, . . . and does not depend on the particular choice of Λ1 . In addition, ˇ˘ `ˇ lim E ˇAΛ ptℓ q ´ AΛ1 Xr1,N s ptℓ qˇ “ 0. N Ñ8. Next let t ě 0 be arbitrary with EpXptqq “ EpXpt´qq “ EpXpt`qq and let Λ1 Ď Λ0 , Λ1 inﬁnitely large. For t1 ă t ă t2 , t1 , t2 in Q1 , we have ˙ ˆ ˙ ˆ 2 E lim sup AΛ1 Xr1,N s ptq ď E lim sup AΛ1 Xr1,N s pt q N Ñ8 N Ñ8 ´ ¯ “ E lim inf AΛ1 Xr1,N s pt2 q ď E pXpt2 q ´ Xp0qq . N Ñ8. Similarly we have ´ ¯ ´ ¯ E lim inf AΛ1 Xr1,N s ptq ě E lim inf AΛ1 Xr1,N s pt1 q N Ñ8 N Ñ8 ˙ ˆ “ E lim sup AΛ1 Xr1,N s pt1 q ě E pXpt1 q ´ Xp0qq . N Ñ8. Since E pXpt`qq “ E pXpt´qq, it follows that the limit AΛ1 ptq :“ lim AΛ1 Xr1,N s ptq N Ñ8. exists P-almost surely. Consequently, the limits AΛ ptq :“ lim AΛ1 Xr1,N s ptq, t ě 0, N Ñ8 ˇ˘ `ˇ all exist P-almost surely and limN Ñ8 E ˇAΛ ptq ´ AΛ1 Xr1,N s ptqˇ “ 0. Finally we shall prove that the process tAΛ ptq : t ě 0u is right continuous. Fix t0 ě 0 and let t ą t0 . Then E pAΛ ptq ´ AΛ pt0 qq “ E pXptq ´ Xpt0 qq ě 0.. 420 Download free eBooks at bookboon.com.

(185) Advanced stochastic processes: Part II. 8. DOOB-MEYER DECOMPOSITION. Some related results 421. Since t ÞÑ EpXptqq is right continuous we infer that limtÓt0 E pAΛ ptq ´ AΛ pt0 qq “ 0. It follows that, P-almost surely, limtÓt0 AΛ ptq “ AΛ pt0 q. This completes the proof of Theorem 5.140.. . 5.141. Corollary. Let X “ M ` A be the Doob-Meyer decomposition of a submartingale into a martingale and an increasing right continuous predictable process A. Then, for an appropriate sequence pnℓ : ℓ P Nq in N, N 8 ˘ ˘ ` ˘˘ 1 ÿ ÿ` ` ` Aptq “ lim E A pj ` 1q2´nk | Fj2´nk ´ A j2´nk 1pj2´nk ,8q ptq. N Ñ8 N k“1 j“0. This limit is an P-almost sure limit as well as a limit in L1 pΩ, F, Pq.. Proof. A combination of the existence and uniqueness of the Doob-Meyer decomposition yields the desired result. Notice that by Proposition 5.131 a process of the form M `A, where M is a martingale and where A is an increasing adapted process in L1 pΩ, F, Pq is of class (DL): see (5.230) and (5.231). So the proof of Corollary 5.141 is complete now. Another corollary is the following one. 5.142. Corollary. Let tXptq : t ě 0u be a right continuous submartingale of class (DL) with left limits. Fix t0 ą 0 and let tτℓ : ℓ P Nu be sequence of stopping times which increases to the fixed time t0 . Suppose τℓ ă t0 , P-almost surely, for all ℓ P N. Then E p|Xpt0 ´q|q ă 8 and limℓÑ8 E p|X pτℓ q ´ X pt0 ´q|q “ 0. In addition, limhÓ0 E p|X pt0 ` hq ´ X pt0 q|q “ 0. The following result also follows from our discussion. 5.143. Corollary. Let tXptq : t ě 0u be a submartingale. If the function t ÞÑ E pXptqq is P-almost surely continuous, then the process tAptq : t ě 0u is Palmost surely continuous as well. 5.144. Remark. Several people have reformulated and extended Komlos’ result as a principle of subsequences, e.g. see Chatterji [30]. Others have treated an inﬁnite dimensional version, e.g. see Balder [7]. In [96], Exercise 3, p. 103 the authors give an example of a submartingale which is not of class (DL). In fact M´etivier and Pellaumail give the following example. Let Ω be the interval r0, 1s with Lebesgue measure and let 0 “ t0 ă t1 ă . . . ă tn ă . . . ă 1 be a sequence such that limnÑ8 tn “ 1. Deﬁne the process X by ÿ8 2n 1rtn´1 ,tn q ptq1p1´2´n ,1s pωq, ω P r0, 1s, t ě 0. Xpt, ωq “ ´ n“1. Then X is a submartingale, X is not of class (DL) and X is a martingale on the interval r0, 1q. If tn´1 ď t ă tn , we write Ft for the σ-ﬁeld generated by tpj ´ 1q2´n , j2n s : 1 ď j ď 2n u. If t ě 1, then Ft is the Borel ﬁeld of r0, 1s.. 5.145. Definition. Let tY ptq : t ě 0u be a martingale in L2 pΩ, F, Pq. Then t|Y ptq|2 : t ě 0u is a submartingale of class (DL). So by Theorem 5.130 there. 421 Download free eBooks at bookboon.com.

(186) Advanced stochastic processes: Part II 422 5. SOME RELATED RESULTS. Some related results. exists a unique martingale tM ptq : t ě 0u with M p0q “ |Y p0q|2 and an increasing predictable right-continuous process t⟨Y ⟩ ptq : t ě 0u in L1 pΩ, F, Pq such that |Y ptq|2 “ M ptq ` ⟨Y ⟩ ptq,. P-almost surely.. The process t⟨Y ⟩ ptq : t ě 0u is called the (quadratic) variation or variance process of tY ptq : t ě 0u.. 5.146. Example. Let tBptq : t ě 0u be ν-dimensional Brownian motion. Then the process tt ÞÑ νt : t ě 0u is the corresponding quadratic variation process. şt şt 5.147. Example. let t ÞÑ 0 F1 psq dBpsq and t ÞÑ 0 F2 psq dBpsq be two local şt martingales. Then the process t ÞÑ 0 F1 psqF2 psq ds is the corresponding covariation process.. Need help with your dissertation? Get in-depth feedback & advice from experts in your topic area. Find out what you can do to improve the quality of your dissertation!. Get Help Now. Go to www.helpmyassignment.co.uk for more info. 422 Download free eBooks at bookboon.com. Click on the ad to read more.

(187) Advanced stochastic processes: II SUBJECTS FOR Part FURTHER RESEARCH AND PRESENTATIONS. Some related results 423. Subjects for further research and presentations The following topics may be of interest for a presentation and/or further research:. (1) Certain pseudo-diﬀerential operators of order less than or equal to 2 can be put into correspondence with space-homogeneous or non-spacehomogeneous Markov processes. A detailed exposition can be found in Jacob [62, 63, 64]. (2) Viscosity solutions to partial diﬀerential equations. The standard reference for this subject is Crandall, Ishii, and Lions [35]. This topic can also be treated in the context of Backward Stochastic Diﬀerential Equations (BSDEs): see, e.g., Pardoux [109]. (3) Elliptic diﬀerential operators of second order (and Markov processes); see, e.g., Øksendael [106]. (4) Parabolic diﬀerential operators (of second order and Markov processes). An interesting article in this context is Bossy and Champagnat [23]. The abstract of this paper reads: “We present the main concepts of the theory of Markov processes: transition semigroups, Feller processes, inﬁnitesimal generator, Kolmogorov’s backward and forward equations, and Feller diﬀusion. We also give several classical examples including stochastic diﬀerential equations (SDEs) and backward stochastic diﬀerential equations (BSDEs) and describe the links between Markov processes and parabolic partial diﬀerential equations (PDEs). In particular, we state the Feynman-Kac formula for linear PDEs and BSDEs, and we give some examples of the correspondence between stochastic control problems and Hamilton-Jacobi-Bellman (HJB) equations and between optimal stopping problems and variational inequalities. Several examples of ﬁnancial applications are given to illustrate each of these results, including European options, Asian options, and American put options.” (5) Solutions to stochastic diﬀerential equations and the corresponding second order diﬀerential equation (of parabolic type) satisﬁed by the onedimensional distributions. (6) Backward stochastic diﬀerential equations and their viscosity solutions; see, e.g. Pardoux [109], Van Casteren [147], Boufoussi and Van Casteren [24, 25], Boufoussi, Van Casteren and Mhardy [26]. (7) Heat equation on a Riemannian manifold. A relevant book in this context is [59]. For connections with stochastic diﬀerential equations on manifolds see, e.g., Elworthy [52, 53]. (8) Oscillatory integrals and related path integrals. There is a lot of literature on this subject. Nice papers on this topic are Albeverio and Mazzucchi [1, 2]. Interesting books are, e.g., Mazzucchi [95], Johnson and Lapidus [65], and Kleinert [69]. (9) Malliavin calculus, or stochastic calculus of variations, and applications to regularity properties of integral kernels. For details see e.g.. 423 Download free eBooks at bookboon.com.

(188) Advanced stochastic processes: 5. Part II 424 SOME RELATED RESULTS. (10). (11). (12). (13) (14) (15). Some related results. Nualart [103, 104]. Other references which contain results on and applications of Malliavin calculus include: Cruzeiro and Malliavin [36], Stroock [127, 128, 129], Cruzeiro and Zambrini [37], [38]. Of course the original work by Malliavin should not be forgotten: [92]. The book by Bismut [18] combines Malliavin calculus with the theory of large deviations. For a discussion on Malliavin calculus in relation to L´evy processes see, e.g., Osswald [108]. A rather elementary approach to Malliavin calculus can be found in Friz [56]. For application to stochastic diﬀerential equations see, e.g., Takeuchi [135]. For applications of Malliavin calculus to operator semigroups see, e.g., L´eandre [83, 84]. For Malliavin calculus without probability theory see [82]. Books and papers with literature on ﬁnancial mathematics include: Le´on, Sol´e, Utzet, and Vives [85], Nualart and Schoutens [105], Malliavin and Thalmaier [93], Karatsas and Shreve [66], Gulisashvili [60], El Karoui and Mazliak [51], El Karoui, Pardoux and Quenez [49], Lim [87]. Other references include Zhang and Zhou (editors) [155] and Tsoi, Nualart and Yin [138]. Another interesting subject is “Ergodic theory” and, correspondingly, invariant measures. We mention some references: Krengel [75], Karlin and Taylor [67], Meyn and Tweedie [97], Eisner and Nagel [48], Van Casteren [146], Seidler [119], Goldys [58], [115]. Central limit theorems and related results are also relevant. Again we mention some references: Bhattaraya and Waymire [15], Nourdin and Peccati [101], Barbour and Chen [8], Berckmoes, Lowen and Van Casteren [11, 12, 13, 14], Tao [137], Stein [124, 125], Chen, Goldstein and Shao [31], Barbour and Hall [9]. Investigate Markov processes with a Polish space as state space: see, e.g., Sharpe [120], Swart and Winter [134], Van Casteren [146], Bovier [27]. Discuss and make a careful study of the Skorohod space as described in Remark 3.40. Try to include applications to convergence properties of stochastic processes. Discuss stochastic analysis in the inﬁnite-dimensional context. A nice and relevant survey paper is [150] written by van Neerven, Veraar and Weis. A simpliﬁed version in Dutch is authored by van Neerven: see [149].. 424 Download free eBooks at bookboon.com.

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(197) Advanced stochastic processes: Part II. Index. Index. Brownian motion, 1, 16–18, 24, 84, 94, 98, 101, 102, 105, 108–110, 113, 115, 181, 189, 193, 197, 243, 283, 290, 291 continuous, 104 distribution of, 107 geometric, 188 H¨older continuity of, 154 pinned, 98 standard, 70 Brownian motion with drift, 98. D: dyadic rational numbers, 380 K: strike price, 191 N p¨q: normal distribution, 191 P 1 pΩq: compact metrizable Hausdorﬀ space, 129 S: spot price, 191 T : maturity time, 191 λ-system, 1, 68 Gδ -set, 332, 334 M: space of complex measures on Rν , 298 µx,y 0,t , 103 π-system, 68 σ-algebra, 1, 3 σ-ﬁeld, 1, 3 σ: volatility, 191 r: risk free interest rate, 191 (DL)-property, 416. cadlag modiﬁcation, 395 cadlag process, 376 Cameron-Martin Girsanov formula, 277 Cameron-Martin transformation, 182, 280 canonical process, 109 Carath´eodory measurable set, 363 Carath´eodory’s extension theorem, 361, 362, 364 central limit theorem, 74 multivariate, 70 Chapman-Kolmogorov identity, 16, 25, 81, 107, 116, 149 characteristic function, 76, 102, 390 characteristic function (Fourier transform), 98 classiﬁcation properties of Markov chains, 35 closed martingale, 17, 150 compact-open topology, 310 complex Radon measure, 296 conditional Brownian bridge measure, 103 conditional expectation, 2, 3, 78 conditional expectation as orthogonal projection, 5 conditional expectation as projection, 5 conditional probability kernel, 399 consistent family of probability spaces, 66 consistent system of probability measures, 13, 360 content, 362 exended, 362 continuity theorem of L´evy, 324. adapted process, 17, 374, 389, 406 additive process, 23, 24 aﬃne function, 8 aﬃne term structure model, 210 Alexandroﬀ compactiﬁcation, 301 almost sure convergence of sub-martingales, 386 arbitrage-free, 190 backward propagator, 197 Banach algebra, 298, 303 Bernoulli distributed random variable, 56 Bernoulli topology, 310 Beurling-Gelfand formula, 302, 303 Birkhoﬀ’s ergodic theorem, 74 birth-dearth process, 35 Black-Scholes model, 187, 190 Black-Scholes parameters, 193 Black-Scholes PDE, 190 Bochner’s theorem, 90, 91, 308, 314 Boolean algebra of subsets, 361 Borel-Cantelli’s lemma, 42, 105 Brownian bridge, 94, 98, 99, 101 Brownian bridge measure conditional, 103 433. 433 Download free eBooks at bookboon.com.

(198) Advanced stochastic processes: Part II 434. contractive operator, 197 convergence in probability, 371, 386 convex function and aﬃne functions, 8 convolution product of measures, 298 convolution semigroup of measures, 314 convolution semigroup of probability measures, 391 coupling argument, 288 covariance matrix, 108, 197, 200, 203 cylinder measure, 360 cylinder set, 358, 367 cylindrical measure, 89, 125 decomposition theorem of Doob-Meyer, 20 delta hedge portfolio, 190 density function, 80 Dirichlet problem, 265 discounted pay-oﬀ, 209 discrete state space, 25 discrete stopping time, 19 dispersion matrix, 94 dissipative operator, 118 distribution of random variable, 102 distributional solution, 266 Dol´eans measure, 168 Donsker’s invariance principle, 71 Doob’s convergence theorem, 17, 18 Doob’s maximal inequality, 21, 23, 160, 384 Doob’s maximality theorem, 21 Doob’s optional sampling theorem, 20, 86, 381, 388, 409 Doob-Meyer decomposition for discrete sub-martingales, 383 Doob-Meyer decomposition theorem, 148, 149, 295, 384, 410, 419, 421 downcrossing, 157 Dynkin system, 1, 68, 111, 300, 378 Elementary renewal theorem, 38 equi-integrable family, 369 ergodic theorem, 295 ergodic theorem in L2 , 342 ergodic theorem of Birkhoﬀ, 76, 340, 344, 354 European call option, 188 European put option, 188 event, 1 exit time, 84 exponential Brownian motion, 186 exponential local martingale, 254, 255 exponential martingale probability measure, 192 extended content, 362 extension theorem. Index. INDEX. of Kolmogorov, 360 exterior measure, 364 face value, 210 Feller semigroup, 79, 113, 114, 120, 121, 140, 264 conservative, 114 generator of, 118, 137, 140, 143, 144 strongly continuous, 113 Feller-Dynkin semigroup, 79, 122, 264 Feynman-Kac formula, 181 ﬁltration, 109, 264 right closure of, 109 ﬁnite partition, 3 ﬁnite-dimensional distribution, 373 ﬁrst hitting time, 18 forward propagator, 197 forward rate, 214 Fourier transform, 90, 93, 96, 102, 251 Fubini’s theorem, 199 full history, 109 function positive-deﬁnite, 305 functional central limit theorem (FCLT), 70, 71 Gaussian kernel, 16, 107 Gaussian process, 89, 110, 115, 200, 203 Gaussian variable, 153 Gaussian vector, 76, 93, 94 GBM, 186 geometric Brownian motion, 189 generator of Feller semigroup, 118, 137, 140, 144, 228, 230, 231, 233 generator of Markov process, 200, 203 geometric Brownian motion, 188 geometric Brownian motion = GBM, 186 Girsanov transformation, 182, 243, 280 Girsanov’s theorem, 193 graph, 232 Gronwall’s inequality, 246 H¨older continuity of Brownian motion, 154 H¨older continuity of processes, 151 Hahn decomposition, 295 Hahn-Kolmogorov’s extension theorem, 364 harmonic function, 86 hedging strategy, 188 Hermite polynomial, 258 Hilbert cube, 333, 334 hitting time, 18 i.i.d. random variables, 24. 434 Download free eBooks at bookboon.com.

(199) Advanced stochastic processes: Part II. index set, 11 indistinguishable processes, 104, 374, 386 information from the future, 374 initial reward, 40 integration by parts formula, 282 interest rate model, 204 internal history, 374, 394 invariant measure, 35, 48, 51, 201, 204 minimal, 50 irreducible Markov chain, 48, 51, 54 Itˆo calculus, 87, 278, 279 Itˆo isometry, 162 Itˆo representation theorem, 274 Itˆo’s lemma, 189, 270 Jensen inequality, 149 Kolmogorov backward equation, 26 Kolmogorov forward equation, 26 Kolmogorov matrix, 26 Kolmogorov’s extension theorem, 13, 17, 89–91, 93, 125, 130, 357, 360, 361, 366 Komlos’ theorem, 295, 409, 420 L´evy’s weak convergence theorem, 115 L´evy process, 89, 389, 390, 392 L´evy’s characterization of Brownian motion, 194, 249 law of random variable, 102 Lebesgue-Stieltjes measure, 364 lemma of Borel-Cantelli, 10, 152 lexicographical ordering, 333 life time, 79, 117 local martingale, 194, 252, 264, 267, 268, 271, 278, 280 local time, 292 locally compact Hausdorﬀ space, 15 marginal distribution, 373 marginal of process, 13 Markov chain, 35, 44, 58, 59, 66 irreducible, 48, 54 recurrent, 48 Markov chain recurrent, 48 Markov process, 1, 16, 29, 30, 61, 79, 89, 102, 110, 113, 115, 119, 144, 202, 406, 408 strong, 119, 406 time-homogeneous, 407 Markov property, 25, 26, 30, 31, 46, 82, 110, 113, 142 strong, 44 martingale, 1, 17, 20, 80–82, 85–88, 103, 109, 243, 280, 281, 378, 382, 396. INDEX. 435. (DL)-property, 227 closed, 17 local, 194 maximal inequality for, 225 martingale measure, 209, 281 martingale problem, 118, 128, 137, 140, 143, 144, 228, 230, 231, 235, 264, 265 uniquely solvable, 118 well-posed, 118 martingale property, 131 martingale representation theorem, 263, 275 maximal ergodic theorem, 351 maximal inequality of Doob, 386 maximal inequality of L´evy, 104 maximal martingale inequality, 225 maximum principle, 118, 140, 141, 143, 232 measurable mapping, 377 measure invariant, 48, 201, 204 mesaure invariant, 204 mesure stationary, 204 metrizable space, 15 Meyer process, 419 minimal invariant measure, 50 modiﬁcation, 374 monotone class theorem, 69, 103, 107, 110, 112, 116, 378, 394, 398, 401, 404 alternative, 378 multiplicative process, 23, 24, 79 multivariate classical central limit theorem, 70 multivariate normal distributed vector, 76 multivariate normally distributed random vector, 93 negative-deﬁnite function, 314, 316, 396 no-arbitrage assumption, 209 non-null recurrent state, 51 non-null state, 47 non-positive recurrent random walk, 57 non-time-homogeneous process, 23 normal cumulative distribution, 188 normal distribution, 197 Novikov condition, 281 Novikov’s condition, 209 null state, 47 num´eraire, 215 number of upcrossings, 156, 379, 380 one-point compactiﬁcation, 15 operator. 435 Download free eBooks at bookboon.com. Index.

(200) Advanced stochastic processes: Part II 436. dissipative, 118 operator which maximally solves the martingale problem, 118, 140, 228 Ornstein-Uhlenbeck process, 98, 102, 200, 201, 210 orthogonal projection, 340 oscillator process, 98, 99 outer measure, 363, 364 partial reward, 40 partition, 4 path, 373 path space, 117 pathwise solutions to SDE, 288, 289 unique, 291, 292 pathwise solutions to SDE’s, 244 payoﬀ process discounted, 193 PDE for bond price in the Vasicek model, 213 pe-measure, 362 persistent state, 47 pinned Brownian motion, 98 Poisson process, 26, 27, 29, 36, 89, 159 Polish space, 15, 90, 123, 334, 335, 360, 361, 366 portfolio delta hedge, 190 positive state, 47 positive-deﬁnite function, 297, 302, 305, 314 positive-deﬁnite matrix, 90, 96, 197 positivity preserving operators, 345 pre-measure, 363, 364 predictable process, 20, 193, 418 probability kernel, 399, 408 probability measure, 1 probability space, 1 process Gaussian, 200, 203 increasing, 21 predictable, 20 process adapted to ﬁltration, 374 process of class (DL), 20, 21, 148, 149, 161, 409–411, 420, 421 progressively measurable process, 377 Prohorov set, 72, 335, 337–339 projective system of probability measures, 13, 121, 360 projective system of probability spaces, 125 propagator backward, 197. Index. INDEX. quadratic covariation process, 249, 264, 279 quadratic variation process, 253 Radon-Nikodym derivative, 11, 408 Radon-Nikodym theorem, 4, 78, 408 random walk, 58 realization, 25, 373 recurrent Markov chain, 48 recurrent state, 47 recurrent symmetric random walk, 55 reference measure, 80, 81, 83 reﬂected Brownian motion, 228 renewal function, 35 renewal process, 35, 40 renewal-reward process, 39, 40 renewal-reward theorem, 41 resolvent family, 122 return time, 55 reward initial, 40 partial, 40 terminal, 40 reward function, 40 Riemann-Stieltjes integral, 364 Riesz representation theorem, 295, 296, 305 right closure of ﬁltration, 109 right-continuous ﬁltration, 374 right-continuous paths, 19 ring of subsets, 361 risk-neutral measure, 193, 209 risk-neutral probability measure, 192 running maximum, 23 sample path, 25 sample path space, 11, 25 sample space, 25 semi-martingale, 419 semi-ring, 364 semi-ring of subsets, 361, 362 semigroup Feller, 264 Feller-Dynkin, 264 shift operator, 109, 117 Skorohod space, 117, 122, 128 Skorohod-Dudley-Wichura representation theorem, 283, 286 Souslin space, 90, 361, 365, 366 space-homogeneous process, 29 spectral radius, 303 standard Brownian motion, 70 state non-null, 47 null, 47. 436 Download free eBooks at bookboon.com.

(201) Advanced stochastic processes: Part II. persistent, 47 positive, 47 recurrent, 47 state space, 11, 17, 79, 117, 400, 406 discrete, 25 state variable, 11, 25, 117 state variables, 125 state:transient, 47 stationary distribution, 25, 51 stationary measure, 204 stationary process, 11 step functions with unit jumps, 159 Stieltjes measure, 364 Stirling’s formula, 54 stochastic diﬀerential equation, 182 stochastic integral, 102, 253 stochastic process, 10 stochastic variable, 11, 371 stochastically continuous process, 159 stochastically equivalent processes, 374 stopped ﬁltration, 377 stopping time, 18, 20, 44, 58, 64, 68, 112, 252, 374–377, 381, 382, 405 discrete, 19 terminal, 18, 24 strong law of large numbers, 41, 76, 155, 340, 344 strong law of large numbers (SLLN), 38 strong Markov process, 102, 119, 121, 140, 406 strong Markov property, 44, 48, 113 strong solution to SDE, 244 strong solutions to SDE unique, 244 strong time-dependent Markov property, 113, 120 strongly continuous Feller semigroup, 113 sub-martingale, 378, 381, 384 sub-probability kernel, 406 sub-probability measure, 1 submartingale, 17, 20, 227 submartingale convergence theorem, 158 submartingale of class (DL), 421 super-martingale, 378 supermartingale, 17, 20 Tanaka’s example, 292 terminal reward, 40 terminal stopping time, 18, 24, 83 theorem Itˆo representation, 274 Kolmogorov’s extension, 278 martingale representation, 275 of Arzela-Ascoli, 72, 73 of Bochner, 90, 304, 308. INDEX. 437. of Doob-Meyer, 20 of Dynkin-Hunt, 397 of Fernique, 221 of Fubini, 199, 330 of Girsanov, 277, 280 of Helly, 334 of Komlos, 409 of L´evy, 253, 270, 290 of Prohorov, 72 of Radon-Nikodym, 290 of Riemann-Lebesgue, 300 of Scheﬀ´e, 39, 278, 369 of Schoenberg, 314 of Stone-Weierstrass, 301, 305 Skorohod-Dudley-Wichura representation, 283, 286 time, 11 time change, 19 stochastic, 19 time-dependent Markov process, 200, 203 time-homogeneous process, 11, 29 time-homogeneous transition probability, 25 time-homogenous Markov process, 407 topology of uniform convergence on compact subsets, 310 tower property of conditional expectation, 5 transient non-symmetric random walk, 57 transient state, 47 transient symmetric random walk, 55 transition function, 119 transition matrix, 51 translation operator, 11, 25, 109, 117, 400, 406 translation variables, 125 uniformly distributed random variable, 394 uniformly integrable family, 5, 6, 20, 39, 369, 388 uniformly integrable martingale, 389 uniformly integrable sequence, 385 unique pathwise solutions to SDE, 244 uniqueness of the Doob-Meyer decomposition, 417 unitary operator, 340, 342 upcrossing inequality, 156, 157, 383 upcrossing times, 156 upcrossings, 156 vague convergence, 371 vague topology, 310, 334 vaguely continuous convolution semigroup of measures, 315. 437 Download free eBooks at bookboon.com. Index.

(202) Advanced stochastic processes: Part II 438. Index. INDEX. vaguely continuous convolution semigroup of probability measures, 389, 390 Vasicek model, 204, 210 volatility, 188 von Neumann’s ergodic theorem, 340 Wald’s equation, 36 weak convergence, 325 weak law of large numbers, 75, 340 weak solutions, 264 weak solutions to SDE’s, 244, 277, 280, 288 unique, 265, 292 weak solutions to stochastic diﬀerential equations, 265 weak topology, 310 weak˚ -topology, 334 weakly compact set, 338, 339 Wiener process, 98. Brain power. By 2020, wind could provide one-tenth of our planet’s electricity needs. Already today, SKF’s innovative knowhow is crucial to running a large proportion of the world’s wind turbines. Up to 25 % of the generating costs relate to maintenance. These can be reduced dramatically thanks to our systems for on-line condition monitoring and automatic lubrication. We help make it more economical to create cleaner, cheaper energy out of thin air. By sharing our experience, expertise, and creativity, industries can boost performance beyond expectations. Therefore we need the best employees who can meet this challenge!. The Power of Knowledge Engineering. Plug into The Power of Knowledge Engineering. Visit us at www.skf.com/knowledge. 438 Download free eBooks at bookboon.com. Click on the ad to read more.

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