▼ô❝ ❧ô❝

✶

❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ❧ý t❤✉②Õt ①➳❝ s✉✃t ✈➭ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥

✸

✶✳✶✳ ❈➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ ♣❤➞♥ ♣❤è✐

✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷✳ ➜é ➤♦ ❤÷✉ ❤➵♥ ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tô❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷✳✶✳ ➜é ➤♦ ❤÷✉ ❤➵♥ ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tô❝ ❜➟♥ tr➳✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷✳✷✳ ➜é ➤♦ ❤÷✉ ❤➵♥ ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tô❝ ❜➟♥ ♣❤➯✐

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✸✳ ➜Þ♥❤ ❧ý ❘❛❞♦♥✲◆✐❝♦❞②♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✹✳ ❈➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❦ú ✈ä♥❣ ❝ã ➤✐Ò✉ ❦✐Ö♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✺✳ ◗✉❛♥ ❤Ö ❣✐÷❛ ❝➳❝ ❦✐Ó✉ ❤é✐ tô

✾

✶✳✺✳✶✳ ◆Õ✉

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

{ξn } ❤é✐ tô ❤➬✉ ❝❤➽❝ ❝❤➽♥ ✈Ò ξ

{ξn } ❤é✐ tô t❤❡♦ ①➳❝ s✉✃t ✈Ò ξ

✳ ✳

✾

✶✳✺✳✷✳ ➜Þ♥❤ ❧ý

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✺✳✸✳ ➜Þ♥❤ ❧ý


✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✺✳✹✳ ◆Õ✉

{ξn } ❤é✐ tô t❤❡♦ ❜×♥❤ ♣❤➢➡♥❣ tr✉♥❣ ❜×♥❤ ✈Ò ξ

s✉✃t ✈Ò
✶✳✺✳✺✳ ◆Õ✉

t❤×

ξ

t❤×

{ξn } ❤é✐ tô t❤❡♦ ①➳❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

{ξn } ❤é✐ tô t❤❡♦ ①➳❝ s✉✃t ✈Ò ξ

{ξn } ❤é✐ tô ②Õ✉ ✈Ò ξ

✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵


✶✳✺✳✻✳ ❚✐➟✉ ❝❤✉➮♥ ❈❛✉❝❤② ✈Ò sù ❤é✐ tô t❤❡♦ ①➳❝ s✉✃t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✺✳✼✳ ❚✐➟✉ ❝❤✉➮♥ ❈❛✉❝❤② ✈Ò sù ❤é✐ tô ❤✳❝✳❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✶✳✼✳ ➜Þ♥❤ ❧ý✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✽✳ ❇æ ➤Ò ❋❛t♦✉✿

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✶✳✾✳ ➜Þ♥❤ ❧ý ❤é✐ tô ❜Þ ❝❤➷♥ ✭▲❡❜❡s❣✉❡✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✶✵✳❈❤ø♥❣ ♠✐♥❤ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳


✶✺

✶✳✻✳ ➜Þ♥❤ ❧ý ❤é✐ tô ➤➡♥ ➤✐Ö✉ ✭❇✳▲❡✈②✮

✶

t❤×

✶✵


❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ▲í♣✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ❑✶✻

✶✳✶✵✳✶✳❇✃t ➤➻♥❣ t❤ø❝ ❍♦❧❞❡r✿

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✶✵✳✷✳❇✃t ➤➻♥❣ t❤ø❝ ▼✐♥❦♦✈s❦✐✿

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✶✳✶✵✳✸✳❇✃t ➤➻♥❣ t❤ø❝ ❏❡♥s❡♥✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✶✳✶✵✳✹✳❇✃t ➤➻♥❣ t❤ø❝ ❈❤❡❜②❡✈✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽


✶✳✶✶✳❇æ ➤Ò ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✭❧✉❐t ✵✲✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✶✳✶✷✳➜Þ♥❤ ❧ý ❋✉❜✐♥✐

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✶✳✶✸✳➜Þ♥❤ ❧ý ✶✳✶✵ ✭❚✐➟✉ ❝❤✉➮♥ ➤ñ ❑♦❧♠♦❣♦r♦✈ ❝❤♦ tÝ♥❤ ❧✐➟♥ tô❝✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✶✳✶✹✳◆Õ✉

✷

✸

✶✺

{ξt }t∈T

❧➭ q✉➳ tr×♥❤ ❣✐❛ sè ➤é❝ ❧❐♣ t❤×

{ξt }t∈T

❧➭ q✉➳ tr×♥❤ ▼❛r❦♦✈ ✳ ✳ ✳ ✳ ✳


✷✶

✶✳✶✺✳❈❤ø♥❣ ♠✐♥❤ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✶✳✶✻✳❱Ý ❞ô ✈Ò t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈

✷✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

▼❛rt✐♥❣❛❧❡ tr➟♥ ❦❤♦➯♥❣ t❤ê✐ ❣✐❛♥ rê✐ r➵❝ ✈➭ ❧✐➟♥ tô❝

✷✹

✷✳✶✳ ▼ét sè ✈Ý ❞ô ✈Ò ▼❛rt✐♥❣❛❧❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

✷✳✷✳ ➜Þ♥❤ ❧ý ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✷✳✸✳ ➜Þ♥❤ ❧ý ✷✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻


✷✳✹✳ ❍Ö q✉➯ ✷✳✺ ✭❜✃t ➤➻♥❣ t❤ø❝ ❑♦❧♠♦❣♦r♦✈✮

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

✷✳✺✳ ❍Ö q✉➯ ✷✳✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

✷✳✻✳ ❍Ö q✉➯ ✷✳✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✷✳✼✳ ▼ét sè ❜➭✐ t❐♣ tr❛♥❣ ✶✹✻ tr♦♥❣ s➳❝❤ ❈➳❝ ♠➠ ❤×♥❤ ①➳❝ s✉✃t ✈➭ ø♥❣ ❞ô♥❣ ✳ ✳ ✳ ✳ ✳

✷✾

◗✉➳ tr×♥❤ ❲✐❡♥❡r ✲ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ■t♦ ✲ P❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥

✸✶

✸✳✶✳ ▼ét sè ❜➭✐ t❐♣ tr❛♥❣ ✶✻✺ tr♦♥❣ s➳❝❤ ❈➳❝ ♠➠ ❤×♥❤ ①➳❝ s✉✃t ✈➭ ø♥❣ ❞ô♥❣ ✳ ✳ ✳ ✳ ✳

✸✶


✸✳✷✳ ●✐➯✐ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✸

✷


❈❤➢➡♥❣ ✶
❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ❧ý t❤✉②Õt ①➳❝
s✉✃t ✈➭ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥

✶✳✶✳

❈➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ ♣❤➞♥ ♣❤è✐

❛✮ ❑❤➠♥❣ ❣✐➯♠✿

Fξ (x1 ) ≤ Fξ (x2 )✱ ✈í✐ x1 ≤ x2 .

❜✮ ▲✐➟♥ tô❝ tr➳✐ tr➟♥

❝✮

R.

Fξ (−∞) = lim Fξ (x) = 0, Fξ (+∞) = lim Fξ (x) = 1
x→−∞

x→+∞


❈❤ø♥❣ ♠✐♥❤✳

❛✮ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤✃t✿ ◆Õ✉
❚❤❐t ✈❐②✱ ❚❛ ❝ã✿
▼➭ râ r➭♥❣ ✈í✐
❉♦ ➤ã✿

❜✮ ❱í✐

t❤×

P (A) ≤ P (B)

B = A ∪ (B \ A). ❉♦ ➤ã✿ P (B) = P (A) + P (B \ A) ≥ P (A)

x1 ≤ x2

t❤×

{ω ∈ Ω : ξ(ω) < x1 } ⊂ {ω ∈ Ω : ξ(ω) < x2 }

Fξ (x1 ) ≤ Fξ (x2 )✱ ✈í✐ x1 ≤ x2 .

x0 ∈ R tï② ý t❛ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ lim− Fξ (x) = Fξ (x0 )

❚❛ ✈✐Õt ❧➵✐
▲✃② ❞➲②
❚❛ t❤✃②
❱×


A⊂B

−1

x→x0
−1

Fξ (x) = P (ξ (−∞, x)) = P ξ (−∞, x)

{xn } t❤á❛ x1 < x2 < · · · < xn < · · · < x0

Bn = (−∞, xn )

✈➭

xn

x0

B0 = (−∞, x0 )

P ξ −1 ❧➭ ➤é ➤♦ ➯♥❤ ✈➭ ➤é ➤♦ ❧➭ ✶ ❤➭♠ ❧✐➟♥ tô❝ ♥➟♥ Fξ (xn ) = P ξ −1 (Bn )

F (x0 )

✸

P ξ −1 (B0 ) =



✹

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

❝✮ ▲✃② ❞➲②
❚❛ t❤✃②
❉♦ ➤ã

{xn } t❤á❛ x1 > x2 > · · · > xn > · · ·

❚ø❝ ❧➭✿

P ξ −1 (∅) = 0

Fξ (−∞) = lim Fξ (x) = 0
x→−∞

❚➢➡♥❣ tù ❧✃② ❞➲②

{xn } t❤á❛ x1 < x2 < · · · < xn < · · ·

Dn = (−∞, xn )

❚❤×

P ξ −1 (Dn )

❚ø❝ ❧➭✿


Fξ (+∞) = lim Fξ (x) = 1

✶✳✷✳✶✳

+∞

P ξ −1 (−∞, +∞) = 1
x→+∞

µ : F → R ❧➭ ➤é ➤♦ ❤÷✉ ❤➵♥

➜é ➤♦ ❤÷✉ ❤➵♥ ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tô❝ ❜➟♥ tr➳✐

❈❤ø♥❣ ♠✐♥❤✳

●✐➯ sö ❝ã ❞➲②

{An } ⊂ F

t❤á❛

A 1 ⊂ A2 ⊂ · · · ⊂ An ⊂ · · ·

A0 = ∅
Bn = An \ An−1 , n = 1, 2, . . .

❚❛ ❝ã✿

xn


➜é ➤♦ ❤÷✉ ❤➵♥ ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tô❝

●✐➯ sö

➜➷t

✈➭

(−∞, +∞)

❉♦ ➤ã✿

✶✳✷✳

−∞

xn

∅

Cn = (−∞, xn )

P ξ −1 (Cn )

✈➭

Bi ∩ Bj = ∅, ∀i = j
∞

∞


Bn =
n=1

An = A
n=1
∞

∞

⇒ µ(A) = µ(

Bn ) =
n=1

µ(Bn )
n=1

n

= lim

n→∞

µ(Bk )
k=1
n

= lim µ
n→∞


Bk
k=1

= lim µ(An )
n→∞

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

✈➭

An

A




ệ ề



ộ ữ ột tụ

ứ

sử

A 1 A2 ã ã ã An ã ã ã


An



A



A=

ó

An =
n=1

An =

Ak
kn

(Ak AC
k+1 )

Ak
kn

kn


à(An ) = à(


à(Ak AC
k+1 )

Ak ) +

kn

k=n


à(Ak AC
k+1 )

= à(A) +
k=n


n

à(Ak AC
k+1 ) < )

à(A) (ì ỗ
n=1



ị ý


sử

P << Q ó tồ t t t ĩ t ố

ớ ợ

0 : P (A) =

dQ, A F
A

ứ


t



ứ sự tồ t



P, Q ộ ữ P 0
K = { : R,

dQ P (A), A F}



A

t

I = sup dQ
K

ó
t

{n } K : lim

n

n dQ = I

n () = max n ()

ó

1kn

n

n K t A F t ó A =

Ak , Ak F



n () = k () tr Ak


k=1
n

n () =

k ()Ak
k=1

n



n dQ =
A

r ọ

n

n

k dQ

k Ak =
A k=1

k=1

Ak


P (Ak ) = P (A)
k=1


✻

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

➜➷t

ξ = sup ξn ≥ ηn

❚❛ t❤✃②✿

ηn

ξ

n
❉♦ ➤ã✿

ηn dQ ≤ P (A), ∀A ∈ F

ξdQ = lim

n→∞

A

A


❈➬♥ ❈▼✿

λ(A) = P (A) −

ξdQ = 0, ∀A ∈ F
A

❚❛ t❤✃②

λ ❧➭ ➤ä ➤♦ ❤÷✉ ❤➵♥ tr➟♥ F

P❤➯♥ ❝❤ø♥❣✱ ●❙

✈➭

λ << Q

∃A ∈ F : λ(A) = 0

+
⇒ ∃n ∈ N, Ω+
n ∈ F, Q(Ωn ) > 0 s❛♦ ❝❤♦✿

1
+
+
Q(A ∩ Ω+
n ) ≤ λ(A ∩ Ωn ) = P (A ∩ Ωn ) −
n


ξdQ, ∀A ∈ F

A∩Ω+
n

❳Ðt

η=ξ+

1
χ + ✳ ❚❛ ❝ã✿
n Ωn
ηdQ =
A

1
Q(A ∩ Ω+
n)
n

ξdQ +
A

≤

ξdQ + P (A ∩ Ω+
n) =

ξdQ −

A∩Ω+
n

A

ξdQ + P (A ∩ Ω+
n)

≤
A\Ω+
n

+
≤ P (A \ Ω+
n ) + P (A ∩ Ωn ) = P (A)

⇒ η ∈ K✳ ❱➠ ❧ý ✈×
ηdQ =
Ω

ξdQ +

1
Q(Ω+
n)
n

Ω

>


sup ξn dQ ≥

ξdQ =

n
Ω

Ω

ξn dQ ≥ lim

> sup
Ω
▼➞✉ t❤✉➱♥ ✈í✐

I = sup ξdQ.
ξ∈K Ω

❱❐②

λ(A) = 0, ∀A ∈ F

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

ξn dQ = I

n→∞

n


Ω


✼

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

•ξ

t❤á❛

ξdQ, ∀A ∈ F

P (A) =

❧➭ ❞✉② ♥❤✃t ✭t❤❡♦ ♥❣❤Ü❛ t➢➡♥❣ ➤➢➡♥❣ ♥❣➱✉ ♥❤✐➟♥✮✳

A
❚❤❐t ✈❐②✱ ●❙

∃ξ1 , ξ2

ξ2 dQ = P (A), ∀A ∈ F

ξ1 dQ =

t❤á❛

A


A

⇔

(ξ1 − ξ2 )dQ = 0 ⇔ ξ1 = ξ2 (h.k.n)
A

✶✳✹✳
❛✮

❈➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❦ú ✈ä♥❣ ❝ã ➤✐Ò✉ ❦✐Ö♥

E(ξ|G) ≥ 0, ∀ξ ≥ 0;

❜✮ ◆Õ✉ ❝ ❧➭ ❤➺♥❣ sè t❤×

❝✮

E(aξ1 + bξ2 |G) = aE(ξ1 |G) + bE(ξ2 |G) ✭P✲❤✳❝✳❝✮❀
G ✲➤♦ ➤➢î❝ ✈➭ ∃E(ξη), ∃E(η) t❤× E(ξη|G) = ξE(η|G) ✭P✲❤✳❝✳❝✮❀

❞✮ ◆Õ✉

ξ

❡✮ ◆Õ✉

G1 ⊂ G2


❢✮ ◆Õ✉

G

❣✮

E(c|G) = c ✭P✲❤✳❝✳❝✮❀

❧➭

✈➭

ξ

t❤×

E[E(ξ|G2 )|G1 ] = E(ξ|G1 ) ✭P✲❤✳❝✳❝✮❀

➤é❝ ❧❐♣ t❤×

E(ξ|G = Eξ

E(ξ|Gmin ) = Eξ, E(ξ|Gmax ) = ξ ❀

✭P✲❤✳❝❝✮❀

✈í✐

Gmin = {∅, Ω}, Gmax = 2Ω .


❈❤ø♥❣ ♠✐♥❤✳

❛✮ ❚❛ ❝ã✿

∀A ∈ G

t❤×

A

E(ξ|G)dP =

A

ξdP ≥ 0

⇒ E(ξ|G) ≥ 0 ✭P✲❤✳❝✳❝✮
❜✮ ❚❛ ❝ã✿

❝✮

∀A ∈ G

A

E(c|G)dP =

A

cdP,


∀A ∈ G ⇒ E(c|G) = c ✭P✲❤✳❝✳❝✮✳

t❛ ❝ã✿

E(aξ1 + bξ2 |G)dP =
A

(aξ1 + bξ2 )dP
A

=a

ξ1 dP + b
A

E(ξ1 |G)dP + b

=a
A

E(ξ2 |G)dP
A

[aE(ξ1 |G) + bE(ξ2 |G)]

=
A

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻


ξ2 dP
A


✽

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

⇒ E(aξ1 + bξ2 |G) = aE(ξ1 |G) + bE(ξ2 |G) ✭P✲❤✳❝✳❝✮
❞✮

• ●✐➯ sö ξ, η ≥ 0 :
n

✯ ❚❍✶✿

xk χAk ✱ tr♦♥❣ ➤ã xk ∈ R, Ak = {ω : ξ(ω) = xk } ⊂ G

ξ=
k=1

❑❤✐ ➤ã✿

∀A ∈ G

t❛ ❝ã✿

n


ξE(η|G)dP =

n

xk χAk E(η|G)dP =

A

A k=1
n

=

E(η|G)dP
A∩Ak

k=1
n

xk

ηdP =
A∩Ak

k=1

n

xk
k=1

n

n

χAk ηdP =
A

xk χAk ηdP
A k=1

E(ξη|G)dP

ξηdP =

=

xk

A k=1

A k=1

⇒ ξE(η|G) = E(ξη|G)(P − h.c.c)
✯ ❚❍✷✿

ξ

❧➭

G−➤♦ ➤➢î❝ ❜✃t ❦ú✳ ❑❤✐ ➤ã✿ ∃{ξn } ❜❐❝ t❤❛♥❣ t❤á❛ ξn → ξ ✳


❚❤❡♦ ❚❍✶ ë tr➟♥ t❤×
▼➭

ξn E(η|G) = E(ξn η|G)

ξn E(η|G) → ξE(η|G) E(ξn η|G) → E(ξη|G)

❉♦ tÝ♥❤ ❞✉② ♥❤✃t ❝ñ❛ ❣✐í✐ ❤➵♥

• ❚r➢ê♥❣ ❤î♣ ξ, η

⇒ ξE(η|G) = E(ξη|G) ✭P✲❤✳❝✳❝✮

tï② ý✿

ξ = ξ+ − ξ−
η = η+ − η−
E(ξη|G) = E[(ξ + − ξ − )(η + − η − )|G)]
= E(ξ + η + |G) − E(ξ + η − |G) − E(ξ − η + |G) + E(ξ − η − |G)
= ξ + E(η + |G) − ξ + E(η − |G) − ξ − E(η + |G) + ξ − E(η − |G)
= ξ + E[(η + − η − )|G] − ξ − E[(η + − η − )|G]
= (ξ + − ξ − )E[(η + − η − )|G]
= ξE(η|G)
❡✮

∀A ∈ G1
A

t❛ ❝ã✿


A ∈ G2

E[E(ξ|G2 )|G1 ]dP =

A

E(ξ|G2 )dP =

⇒ E[E(ξ|G2 )|G1 ] = E(ξ|G1 ) ✭P✲❤✳❝✳❝✮
❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

A

ξdP =

A

E(ξ|G1 )dP


✾

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

G

❢✮ ❱×

✈➭


❉♦ ➤ã✿

= Eξ

ξ

A
Ω

➤é❝ ❧❐♣ ♥➟♥

E(ξ|G)dP =

χA dP = Eξ

⇒ E(ξ|G) = Eξ
❣✮ ❱í✐

A ∈ Gmin

A

✶✳✺✳

❧➭

dP =

A


✈➭

A

ξχA dP = E(ξχA ) = EξEχA

χA

➤é❝ ❧❐♣✳

EξdP

E(ξ|Gmin )dP =
E(ξ|Gmin )dP =

A

⇒ E(ξ|Gmin ) = Eξ

ξ

ξdP =

ξ

A = ∅ ❤♦➷❝ A = Ω

t❤×


• ❱í✐ A = Ω t❤×✿

∀A ∈ Gmax

A

A

t❤×

✭P✲❤✳❝✳❝✮

• ❱í✐ A = ∅ t❤×✿

❱×

∀A ∈ G

t❛ ❝ã✿

A
A

ξdP = 0 =

A

ξdP = Eξ =

EξdP

A

EξdP

✭P✲❤✳❝✳❝✮

E(ξ|Gmax )dP =

A

A

ξdP

Gmax − ➤♦ ➤➢î❝ ♥➟♥ E(ξ|Gmax ) = ξ

✭P✲❤✳❝✳❝✮

◗✉❛♥ ❤Ö ❣✐÷❛ ❝➳❝ ❦✐Ó✉ ❤é✐ tô

{ξn } ❧➭ ❞➲② ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥✱ ξ
✶✳✺✳✶✳

◆Õ✉

❧➭ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥

{ξn } ❤é✐ tô ❤➬✉ ❝❤➽❝ ❝❤➽♥ ✈Ò ξ

t❤×


{ξn } ❤é✐ tô t❤❡♦ ①➳❝ s✉✃t ✈Ò

ξ
∞
❚❛ ❝ã✿

❈❤ø♥❣ ♠✐♥❤✳

k=n

∞

{|ξk − ξ| ≥ }

An =

➜➷t

{|ξk − ξ| ≥ }] → 0, ∀ > 0

P[

k=n

Bn = {|ξn − ξ| ≥ } ⊂ An

❚❛ t❤✃②
❉♦ ➤ã✿


0 ≤ P (Bn ) ≤ P (An ) → 0

⇒ lim Bn = 0.
n→∞

❱❐②

P

ξn −
→ ξ.

◆♦t❡✳ ➜✐Ò✉ ♥❣➢î❝ ❧➵✐ ♥ã✐ ❝❤✉♥❣ ❦❤➠♥❣ ➤ó♥❣✳

✶✳✺✳✷✳

◆Õ✉

➜Þ♥❤ ❧ý

P

ξn −
→ξ

✈➭

ξn

( ) t❤× ξn → ξ ✭P✲❤✳❝✳❝✮


❚❤❐t ✈❐②✱ ❦❤➠♥❣ ♠✃t tÝ♥❤ tæ♥❣ q✉➳t t❛ ①Ðt ❞➲② ξn

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

P

,−
→ 0 ✭✈× ♥Õ✉ ξn

P

,−
→ ξ t❤× ξn −ξ

P

,−
→ 0✮


✶✵

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

⇒ ∃ > 0 : P{

0(P − h.c.c)

ξn


P❤➯♥ ❝❤ø♥❣✱ ❣✐➯ sö

{ξk ≥ }} > δ > 0
k≥n

❱×

{ξn }

{ξk ≥ } ⊂ {ξn ≥ }

♥➟♥

k≥n
❉♦ ➤ã✿

P

P {ξn ≥ } ≥ P (

{ξk ≥ }) > δ > 0 ✭▼➞✉ t❤✉➱♥ ✈í✐ ξn −
→ ξ✮

k≥n

✶✳✺✳✸✳

◆Õ✉


➜Þ♥❤ ❧ý

P

ξn −
→ξ

t❤×

∃{ξnk } ⊂ {ξn } : ξnk → ξ(P − h.c.c)
∞

{ n}

❚❤❐t ✈❐②✿ ▲✃②

0; {δn } :

δn < ∞
n=1

❱×

P

ξn −
→ξ

♥➟♥ t❛ ❝❤ä♥ ➤➢î❝


nk

s❛♦ ❝❤♦✿

P {|ξnk − ξ| ≥

k}

≤ δk

∞
➜➷t

{|ξnk − ξ| ≥

Rj =

k}

k=j
∞

∞

⇒ P[

∞

{|ξnk − ξ| ≥


k}

{|ξnk − ξ| ≥

= lim P (
j→∞

j=1 k=j
∞

∞

≤ lim

j→∞ k=j

∞

k}

k=j

P {|ξnk − ξ| ≥

k}

≤ lim

j→∞ k=j


δk = 0

∞

{|ξnk − ξ| ≥

⇒ P[

k}

= 0✳ ❚ø❝ ❧➭ ξnk → ξ(P − h.c.c)

j=1 k=j

✶✳✺✳✹✳

◆Õ✉

{ξn }

❤é✐ tô t❤❡♦ ❜×♥❤ ♣❤➢➡♥❣ tr✉♥❣ ❜×♥❤ ✈Ò

t❤❡♦ ①➳❝ s✉✃t ✈Ò

❈❤ø♥❣ ♠✐♥❤✳

●❙

ξ


t❤×

{ξn }

ξ
n→∞

ξ = l.i.mn→∞ ξn ✳ ❚ø❝ ❧➭ E(|ξn − ξ|2 ) =

|ξn − ξ|2 dP −−−→ 0
Ω

❱í✐

tï② ý✱ ➤➷t

❑❤✐ ➤ã✿

Bn = {ω : |ξn − ξ| ≥ }

E(|ξn − ξ|2 ) =
≥

Bn

Bn

|ξn − ξ|2 dP +

|ξn − ξ|2 dP ≥


Bn C

2
Bn

|ξn − ξ|2 dP

dP =

2

P (Bn )

⇒ lim Bn = 0
n→∞

❱❐②

P

ξn −
→ξ

✶✳✺✳✺✳

◆Õ✉

❈❤ø♥❣ ♠✐♥❤✳


❱í✐

{ξn } ❤é✐ tô t❤❡♦ ①➳❝ s✉✃t ✈Ò ξ

❚❛ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤

x , x ∈ R t❤á❛ x < x < x

t❤×

{ξn } ❤é✐ tô ②Õ✉ ✈Ò ξ

Fn (x) → F (x), ∀x ∈ C(F )

✳ ❚❛ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤

F (x ) ≤ lim Fn (x) ≤ lim Fn (x) ≤ F (x )
❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

(∗)

❤é✐ tô




ệ ề

ừ


x

x t ợ Fn (x) F (x)

x, x

ứ ợ t

F (x ) = P { < x } = P { < x ; n < x} + P { < x ; n x}
P {n < x} + P { < x ; n x}
= Fn (x) + P { < x ; n x}
t
r

P

lim P { < x ; n x} lim P {|n | < x x } = 0 n


F (x ) lim Fn (x)

tự t ét

Fn (x) = P {n < x} = P {n < x; < x } + P {n < x; x }
F (x ) + P {|n | > x x }

lim Fn (x) F (x )


ề sự ộ tụ t st


ị ĩ

{n } ợ ọ t st ế > 0
P {|n m | > } 0 n, m

ị ý

{n } ộ tụ t st ỉ ó t

st

ứ

ề ệ sử

P

n
ó > 0 t ó
n,m

P {|n m | > } P {|n | > } + P {|m | > } 0
2
2
ề ệ ủ từ ở ột ết q ế

tồ t

{n }


t st tì

{nk } ộ tụ t st ế ế ó ừ t tứ

P {|n | > } P {|n nk | > } + P {|nk | > }
2
2


n, nk t s r ề ứ

r ọ


✶✷

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

✶✳✺✳✼✳

❚✐➟✉ ❝❤✉➮♥ ❈❛✉❝❤② ✈Ò sù ❤é✐ tô ❤✳❝✳❝

{ξn } ➤➢î❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② P✲❤✳❝✳❝ ♥Õ✉✿ ∀ > 0

➜Þ♥❤ ♥❣❤Ü❛✳ ❉➲②

P { sup |ξk − ξl | ≥ } → 0 ❦❤✐ n → ∞
k,l≥n


➜Þ♥❤ ❧ý✳ ❉➲②

{ξn }

❤é✐ tô P✲❤✳❝✳❝ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

{ξn }

❧➭ ❞➲② ❈❛✉❝❤② t❤❡♦ ♥❣❤Ü❛

P✲❤✳❝✳❝

❈❤ø♥❣ ♠✐♥❤✳

➜✐Ò✉ ❦✐Ö♥ ❝➬♥✳ ●✐➯ sö

h.c.c

ξn −−→ ξ ✳

❚❛ ❝ã✿

P { sup |ξk − ξl | ≥ } ≤ P {sup |ξk − ξ| ≥ } + P {sup |ξl − ξ| ≥ }
2
2
k,l≥n
k≥n
l≥n
❱×


h.c.c

ξn −−→ ξ

n→∞

n→∞

P {sup |ξk − ξ| ≥ 2 } −−−→ 0 ✈➭ P {sup |ξl − ξ| ≥ 2 } −−−→ 0

♥➟♥

k≥n

❙✉② r❛

l≥n

n→∞

P { sup |ξk − ξl | ≥ } −−−→ 0
k,l≥n

➜✐Ò✉ ❦✐Ö♥ ➤ñ✳ ●✐➯ sö

{ξn } ❧➭ ❞➲② ❈❛✉❝❤② t❤❡♦ ♥❣❤Ü❛ P✲❤✳❝✳❝✳

❚❛ ❝ã ♠ét ❦Õt q✉➯✿ ◆Õ✉

{ξn }


❧➭ ❞➲② ❈❛✉❝❤② t❤❡♦ ♥❣❤Ü❛ P✲❤✳❝✳❝ t❤× ✈í✐ ①➳❝ s✉✃t ✶✱ ❝➳❝

{ξn (ω)} ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ R
❉♦ ➤ã✿

˜
ξn (ω) → ξ(ω)
♥➭♦ ➤ã✳ ➜➷t
ξ(ω) =

❑❤✐ ➤ã✿

✶✳✻✳
◆Õ✉

E(ξn |G)

˜
ξ(ω)
0

ω
t➵✐ ω
t➵✐

♠➭ ❣✐í✐ ❤➵♥ tå♥ t➵✐
♠➭ ❣✐í✐ ❤➵♥ ❦❤➠♥❣ tå♥ t➵✐

h.c.c


ξn −−→ ξ

➜Þ♥❤ ❧ý ❤é✐ tô ➤➡♥ ➤✐Ö✉ ✭❇✳▲❡✈②✮

G ⊂ F, ξn

ξ

✭P✲❤✳❝✳❝✮ ✈➭

E(ξ|G) ✭❤♦➷❝ E(ξn |G)

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

M ξ1− < ∞

✭❤♦➷❝

ξn

E(ξ|G) t➢➡♥❣ ø♥❣✮

ξ

✭P✲❤✳❝✳❝✮ ✈➭

M ξ1+ < ∞✮

t❤×



✶✸

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

•

❈❤ø♥❣ ♠✐♥❤✳

❚❛ ❝ã✿

●✐➯ sö

0 ≤ ξn + ξ1−

ξn

ξ

✈➭

M ξ1− < ∞

ξ + ξ1− , ∀n ≥ 1

ξ1 ≤ ξ2 ≤ · · · ξn ≤ · · ·

❚❤❐t ✈❐②✱ ❱×


✈➭

ξ1− = max{−ξ1 , 0} ♥➟♥

✰✮ ❱í✐

ξ1− = 0 ⇒ ξ1 ≥ 0 ⇒ ξn + ξ1− = ξn ≥ ξ1 ≥ 0, ∀n ≥ 1

✰✮ ❱í✐

ξ1− = −ξ1 ⇒ ξ1 < 0 ⇒ ξn + ξ1− = ξn − ξ1 ≥ 0, ∀n ≥ 1

∀A ∈ G

t❛ ❝ã✿

A

[ lim E(ξn |G) + E(ξ1− |G)]dP

E(ξ1− |G)]dP =

lim E(ξn |G) +

n→∞

n→∞

A


A

lim E[(ξn + ξ1− )|G]dP

=

n→∞
A

=

E[(ξn + ξ1− )|G]dP

lim

n→∞
A

=

(ξn + ξ1− )dP

lim

n→∞

A

(ξ + ξ1− )dP


=
A

E[(ξ + ξ1− )|G]dP

=
A

E(ξ1− |G)dP

E(ξ|G)dP +

=
A

A

⇒ lim E(ξn |G) = E(ξ|G) ✭P✲❤✳❝✳❝✮
n→∞

•
✶✳✼✳

❚r➢ê♥❣ ❤î♣

ξn

ξ

✭P✲❤✳❝✳❝✮ ✈➭


M ξ1+ < ∞✮ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù✳

➜Þ♥❤ ❧ý✿

{ξα }α∈U

❦❤➯ tÝ❝❤ ➤Ò✉ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

∃η ≥ 0 ❦❤➯ tÝ❝❤✿ |ξα | ≤ η

❈❤ø♥❣ ♠✐♥❤✳
➜✐Ò✉ ❦✐Ö♥ ❝➬♥✳ ●✐➯ sö

❑❤✐ ➤ã✱ t❛ ❝ã

{ξα }α∈U

❦❤➯ tÝ❝❤ ➤Ò✉✳

sup E|ξα | < +∞✳ ❉♦ ➤ã✱ t❛ ❝❤ä♥ η = ξα0
α

➜✐Ò✉ ❦✐Ö♥ ➤ñ✳ ●✐➯ sö

❚❛ ❝ã✿

∃η ≥ 0 ❦❤➯ tÝ❝❤✿ |ξα | ≤ η ∈ L

|η(ω)| ≥ |ξα (ω)| > x


❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

♠➭

Eα0 = sup E|ξα | ❧➭ t❤á❛✳
α


❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

✶✹

⇒ {ω : |ξα (ω)| > x} ⊂ {ω : |η(ω)| > x}
⇒ sup

x→∞

|ξα |dP ≤

α∈U {|ξα |>x}

|η|dP −−−→ 0

✭❱×

{|η| > x} → ∅ ❦❤✐ x → ∞✮

{|η|>x}
x→∞


⇒ sup

|ξα |dP −−−→ 0

α∈U {|ξα |>x}

{ξα }α∈U

❱❐②

✶✳✽✳

❦❤➯ tÝ❝❤ ➤Ò✉✳

❇æ ➤Ò ❋❛t♦✉✿

{ξn }n≥1

◆Õ✉ ❞➲②

❦❤➯ tÝ❝❤ ➤Ò✉ t❤×✿

❛✮

E(lim ξn |G) ≤ lim E(ξn |G)

❜✮

E(lim ξn |G) ≥ lim E(ξn |G)

❈❤ø♥❣ ♠✐♥❤✳

❛✮ ❚❛ ❝ã✿
❱×

inf ξm

m≥n

{ξn }n≥1

lim ξn

✭✯✮

❦❤➯ tÝ❝❤ ➤Ò✉ ♥➟♥

∃η ≥ 0 ❦❤➯ tÝ❝❤✿ |ξn | ≤ η ⇔ −η ≤ ξn ≤ η

ξn ≥ −η ⇒ ξn− ≤ (−η)− = η ✳ ❉♦ ➤ã✿ ( inf ξm )− ≤ (−η)−

❱í✐

m≥n

−

⇒ E( inf ξm ) ≤ Eη < +∞

✭✯✯✮


m≥n

❚õ ✭✯✮ ✈➭ ✭✯✯✮✱ t❤❡♦ ➤Þ♥❤ ❧ý ❇✳▲❡✈② t❛ ❝ã✿

lim E( inf ξm |G) = E(lim ξn |G)
n

▼➭

m≥n

lim E( inf ξm |G) ≤ lim E(ξn |G)

❱❐②

n

m≥n

n

E(lim ξn |G) ≤ lim E(ξn |G)

❜✮ ❚➢➡♥❣ tù t❛ ❝ò♥❣ ❝ã✿

sup ξm

lim ξn


m≥n

(sup ξm )+ ≤ η + = η
⇒ E(sup ξm )+ ≤ Eη < +∞
m≥n

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻


✶✺

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

⇒ lim E(sup ξm |G) = E(lim ξn |G)
m≥n
❱➭

lim E(sup ξm |G) ≥ lim E(ξn |G)
m≥n

❱❐②

✶✳✾✳

E(lim ξn |G) ≥ lim E(ξn |G)

➜Þ♥❤ ❧ý ❤é✐ tô ❜Þ ❝❤➷♥ ✭▲❡❜❡s❣✉❡✮

◆Õ✉ ❞➲②


ξn → ξ

❈❤ø♥❣ ♠✐♥❤✳

➜➷t

✭P✲❤✳❝✳❝✮ ✈➭

∃η ∈ L1 : |ξn | ≤ η

t❤×

h.c.c

E(|ξn − ξ||G) −−→ 0✳

Yn = sup |ξm − ξ|
m≥n

❚❛ t❤✃②✿

Yn

0

0 ≤ Yn ≤ 2η
⇒0≤

E(Yn |G)dP =
Ω


Yn dP → 0
Ω

⇒ E(Yn |G) → 0 ✭P✲❤✳❝✳❝✮
▼➷t ❦❤➳❝✿

|ξn − ξ| ≤ sup |ξm − ξ| = Yn
m≥n

❱❐②

E(|ξn − ξ||G) → 0 ✭P✲❤✳❝✳❝✮✳

✶✳✶✵✳

❈❤ø♥❣ ♠✐♥❤ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝

❈❤♦ ❝➳❝ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥

✶✳✶✵✳✶✳

ξ, η ✳

❇✃t ➤➻♥❣ t❤ø❝ ❍♦❧❞❡r✿

E(|ξη|) ≤ ξ p . η q , ∀p, q > 1 :

1 1
+ = 1,

p q

✈í✐

ξ

1

p

= [E(|ξ|p )] p

• ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ ❜✃t ➤➻♥❣ t❤ø❝ s❛✉✿
1
1
a b
1 1
+ ≥ a p .b q ; a, b > 0, p, q > 1 : + = 1
p q
p q
p
❚❤❐t ✈❐②✱ t❛ t❤✃② ❤➭♠ f (x) = x , ✈í✐ p > 1 ❧➭ ❤➭♠ ❧å✐ tr➟♥ (0, +∞)

(1)

❈❤ø♥❣ ♠✐♥❤✳

(1.1)

⇒ f (x) − f (1) ≥ f (1)(x − 1), ∀x > 0

⇔ xp − 1 ≥ p(x − 1)
a p1
❚❤❛② x = ( ) , a, b > 0 ✈➭♦ ✭✶✳✷✮ t❛ ➤➢î❝✿
b

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

(1.2)


✶✻

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

a
a p1
− 1 ≥ p[( ) − 1]
b
b
⇔

1
1
1
1
1
1
a b
a
1

a b
1
1
− ≥ a p .b1− p − b ⇔ + (1 − )b ≥ a p .b1− p ⇔ + ≥ a p .b q (➜➷t = 1 − )
p p
p
p
p q
q
p

• ❈❤ø♥❣ ♠✐♥❤ ❜✃t ➤➻♥❣ t❤ø❝ ❍♦❧❞❡r✿
ξ p. η

✰✮ ●✐➯ sö

q

=0:
(1) ⇔
|ξ|
ξ p

➳♣ ❞ô♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✶✮ ❝❤♦ a =
1
p
⇒
❍❛②

ξ p. η


✰✮ ◆Õ✉

q

|ξ|
ξ p

p

1
+
q

E|ξη|
ξ p. η

≤1
q

p

|η|
η q

,b =
|η|
η q

q


≥

q
t❛ ➤➢î❝ ❦Õt q✉➯✿

|ξ|.|η|
ξ p. η

E|ξ.η|
1 E(|ξ|p ) 1 E(|η|q )
p +
q ≥
p ( ξ p)
q ( η q)
ξ p. η

1=

q

q

E|ξ.η|
1 E(|ξ|p ) 1 E(|η|q )
+
≥
p
q
p E(|ξ| ) q E(|η| )

ξ p. η

q

= 0 ⇔ E(|ξ|p )E(|η|q ) = 0

❑❤➠♥❣ ♠✃t tÝ♥❤ tæ♥❣ q✉➳t t❛ ❣✐➯ sö

E(|ξ|p ) = 0

⇒ ξ = 0(P − h.c.c) ⇒ E|ξ.η| = 0(P − h.c.c) ✭▲ó❝ ➤ã ①➯② r❛ ➤➻♥❣ t❤ø❝✮
✶✳✶✵✳✷✳

ξ+η

❇✃t ➤➻♥❣ t❤ø❝ ▼✐♥❦♦✈s❦✐✿

p

≤ ξ

❈❤ø♥❣ ♠✐♥❤✳

p

+ η p , ∀p ≥ 1.

(2)

❚r➢í❝ ❤Õt t❛ ❝ã ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ✭sÏ ❝❤ø♥❣ ♠✐♥❤ s❛✉✮✿


(a + b)p ≤ 2p−1 (ap + bp ), ∀a, b > 0, p ≥ 1
❚r♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✶✮✱ ❧✃②

a = |ξ|, b = |η| t❛ ➤➢î❝✿

|ξ + η|p ≤ (|ξ| + |η|)p ≤ 2p−1 (|ξ|p + |η|p )
• ❱í✐ p = 1 t❤× (2.2) ⇒ E|ξ + η| ≤ E(|ξ|p ) + E(|η|p ) ✭❝❤Ý♥❤ ❧➭ ✭✷✮ tr♦♥❣ ❚❍ ♣ ❂✶✮
• ❱í✐ p > 1 :
|ξ + η|p = |ξ + η|.|ξ + η|p−1 ≤ |ξ||ξ + η|p−1 + |η||ξ + η|p−1

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

✭✷✳✶✮

✭✷✳✷✮


✶✼

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

⇒ E|ξ + η|p ≤ E(|ξ||ξ + η|p−1 ) + E(|η||ξ + η|p−1 )
1 1
+ = 1 ✈➭ ➳♣ ❞ô♥❣ ❜➤t ❍♦❧❞❡r t❛ ➤➢î❝✿
▲✃② q > 1 :
p q
1

(2.3)


1

E(|ξ||ξ + η|p−1 ) ≤ (E|ξ|p ]) p .(E|ξ + η|(p−1)q ) q =
1
1
1 1
≤ (E|ξ|p ) p .(E|ξ + η|p ) q = ( ✈× + = 1 ⇔ p = (p − 1)q)
p q
1 p

1

1

≤ (E|ξ|p ) p .(E|ξ + η|p ) p . q = ξ p ( ξ + η pp ) q
1

❚➢➡♥❣ tù✿
❱➭

E(|η||ξ + η|p−1 ) ≤ η p ( ξ + η pp ) q

E|ξ + η|p = ξ + η

p
p

❚❤❛② ✈➭♦ ✭✷✳✸✮✱ t❛ ➤➢î❝✿


ξ+η
⇔

ξ+η

p
p

p
p

≤( ξ

1− 1q

❍❛②

1

p

≤ ξ

ξ+η

+ η p )( ξ + η pp ) q
p

p


+ η

≤ ξ

p

p

+ η

(1 −

1
1
= )
q
p

p

❈✉è✐ ❝ï♥❣ t❛ ❈▼ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✶✮ ë tr➟♥

❳Ðt ❤➭♠ sè

f (x) = (a + x)p − 2p−1 (ap + xp ), x > 0

❉Ô ❞➭♥❣ t❤✃②

f (x) ≤ f (a) = 0, ∀x > 0


❉♦ ➤ã ✈í✐

x = b t❤× (a + b)p ≤ 2p−1 (ap + bp )

✶✳✶✵✳✸✳

❇✃t ➤➻♥❣ t❤ø❝ ❏❡♥s❡♥✿

❈❤♦ ❤➭♠

f : R −→ R ❧å✐✱ ξ ∈ L1 , E(|f (ξ)|) < ∞.❑❤✐ ➤ã✿
f (Eξ) ≤ Ef (ξ).

❈❤ø♥❣ ♠✐♥❤✳

❱× ❢ ❧➭ ❤➭♠ ❧å✐ ♥➟♥ t❛ ❝ã

k(x0 ) =
▲✃②

x0 = Eξ, x = ξ

f (x) − f (x0 ) ≥ k(x0 )(x − x0 )✱ tr♦♥❣ ➤ã✿
f (x0 − )
f (x0 + )

∃f (x0 − )
+
♥Õ✉ ∃f (x0 )


♥Õ✉

t❤× t❛ ➤➢î❝✿

f (ξ) − f (Eξ) ≥ k(Eξ)(ξ − Eξ)
⇒ E[f (ξ) − f (Eξ)] ≥ k(Eξ)E(ξ − Eξ) = 0
❱❐②

Ef (ξ) ≥ f (Eξ)

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻


✶✽

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

✶✳✶✵✳✹✳

❇✃t ➤➻♥❣ t❤ø❝ ❈❤❡❜②❡✈✿

P {|ξ| > a} ≤

E|ξ|
, ∀ξ ∈ L1 , ∀a > 0
a

❈❤ø♥❣ ♠✐♥❤✳

❚❛ ❝ã✿


|ξ|dP =

E|ξ| =
Ω

⇒ P {|ξ| > a} ≤

|ξ|dP +
{|ξ|>a}

{|ξ|≤a}

Ef |ξ| =

Ef |ξ|
, ∀ξ ∈ L1 , ∀a > 0
f (a)

f |ξ|dP +
{|ξ|>a}

✶✳✶✶✳

{|ξ|>a}

f : R+ −→ R+ ✱ ❦❤➠♥❣ ❣✐➯♠ t❤×
P {|ξ| > a} ≤

P {|ξ| > a} ≤


|ξ|dP ≥ a.P {|ξ| > a}

E|ξ|
a

❚æ♥❣ q✉➳t✿ ◆Õ✉

❚❤❐t ✈❐②✱

|ξ|dP ≥

f |ξ|dP ≥
{|ξ|≤a}

f |ξ|dP ≥ f (a).P {|ξ| > a} ⇒
{|ξ|>a}

Ef |ξ|
f (a)

❇æ ➤Ò ❇♦r❡❧✲❈❛♥t❡❧❧✐ ✭❧✉❐t ✵✲✶✮

❈❤♦ ❞➲② ❝➳❝ ❜✐Õ♥ ❝è

{An }n≥1 ⊂ F, A∗ := lim An :=

∞

∞


Am ✳ ❑❤✐ ➤ã✿
n=1 m=n

∞
❛✮

P (An ) < ∞ ⇒ P (A∗ ) = 0.

n=1
∞
❜✮ ◆Õ✉ t❤➟♠ ❣✐➯ t❤✐Õt ❞➲②

{An }n≥1

➤é❝ ❧❐♣ t❤×✿

P (An ) = ∞ ⇒ P (A∗ ) = 1.

n=1

❈❤ø♥❣ ♠✐♥❤✳

∞
❛✮ ❚❛ ❝ã

Am }n≥1

{
m=n


P (A∗ ) = lim P (
n→∞

❧➭ ❞➲② ❣✐➯♠ ♥➟♥

∞

m=n

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

∞

Am ) ≤ lim

n→∞ m=n

∞

P (An ) < ∞✮

P (Am ) = 0 ✭❱×
n=1


✶✾

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆


{An }n≥1

❜✮ ●✐➯ sö

⇒ P(

⇒ {An }n≥1

➤é❝ ❧❐♣

➤é❝ ❧❐♣

∞

∞

Am ) =

P (Am ) ❉♦ ➤ã t❛ ❝ã✿
m=n

m=n

∞

∞

0 ≤ P(

∞


Am ) =

(1 − P (Am ))

P (Am ) =
m=n

m=n

m=n
∞

∞
−P (Am )

≤

e

−

(∗) = e

P (Am )
m=n

= e−∞ = 0

m=n

∞

⇒ P(

∞

m=n
✭✭✯✮ sö ❞ô♥❣ ❜✃t ➤➻♥❣ t❤ø❝

✶✳✶✷✳

m=n
−x

1 − x ≤ e , 0 ≤ x ≤ 1✮

➜Þ♥❤ ❧ý ❋✉❜✐♥✐

{ξt }t∈T

❈❤♦ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥

❛✮

Am ) = 1 ⇒ P (A∗ ) = 1

Am ) = 0 ❤❛② P (

ξ(t, ω) ➤♦ ➤➢î❝ t❤❡♦ t ∈ T


✭P✲❤✳❝✳❝✮❀

❜✮ ◆Õ✉

∃Eξt , ∀t t❤× mt := Eξt

❝✮ ◆Õ✉

S

➤♦ ➤➢î❝ tr➟♥

➤♦ ➤➢î❝✳ ❑❤✐ ➤ã✿

➤♦ ➤➢î❝ t❤❡♦ t❀

E|ξt |dt < ∞ t❤×✿

T = [0, +∞) ✈➭
S



 |ξt |dt < ∞(P − h.c.c)
S


 E|ξt |dt = E
S


|ξt |dt
S

❈❤ø♥❣ ♠✐♥❤✳

❛✮ ❚❛ ❝ã✿

{(t, ω) ∈ T × Ω : ξ(t, ω) ∈ B} ∈ BT × F, ∀B ∈ B

❉♦ ➤ã✱ ✈í✐ ♠ç✐

ω

❝è ➤Þ♥❤ t❤×

{t ∈ T : ξ(t, ω) ∈ B} ∈ BT , ∀B ∈ B

⇒ ξ(•, ω) : T −→ R ➤♦ ➤➢î❝ ✭t❤❡♦ t✮✳
❜✮

m : T → R, t → mt := Eξt

∀B ∈ B : {t ∈ T : mt ∈ B} = {t ∈ T :

ξt dP ∈ B} ∈ B
Ω

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻



✷✵

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

❝✮

• ❈▼

|ξt |dt < ∞ ✭P✲❤✳❝✳❝✮

✭✸✳✶✮

S

E|ξt |dt < ∞ ⇒ E|ξt | < ∞ ✭P✲❤✳❝✳❝✮
S

⇒ |ξ| < ∞ ✭P✲❤✳❝✳❝✮✱ ∀t ∈ S, ω ∈ Ω
⇒

|ξt |dt < ∞ ✭P✲❤✳❝✳❝✮
S

• ❈▼

E|ξt |dt = E
S

✭❱×


S

|ξt |dt

✭✸✳✷✮

S

➤♦ ➤➢î❝ tr➟♥

P❤➞♥ ❤♦➵❝❤ ➤♦➵♥

T = [0, +∞) ♥➟♥ S = [a, b]✱ ✈í✐ 0 ≤ a ≤ b < +∞

[a, b] t❤➭♥❤ n ➤♦➵♥ ♥❤á✿

a = x0 < x1 = a + h < · · · < xn = a + nh = b,

✈í✐

b−a
n

h=

|ξt |dt < ∞(P − h.c.c) ♥➟♥✿

❱×

S


In =

n
n→∞

ξ(a + ih, •) −−−→

ξt dt
S

i=1

E|ξt |dt < ∞ ♥➟♥✿

▲➵✐ ❝ã✿

S

n
1
Eξ(a
n→∞ h i=1

Eξt dt = lim
S

✶✳✶✸✳

1

h

+ ih, •) = lim EIn = E lim In = E
n→∞

n→∞

ξt dt
S

➜Þ♥❤ ❧ý ✶✳✶✵ ✭❚✐➟✉ ❝❤✉➮♥ ➤ñ ❑♦❧♠♦❣♦r♦✈ ❝❤♦ tÝ♥❤ ❧✐➟♥
tô❝✮

◆Õ✉ ✈í✐

T = [a, b], ∃α > 0, > 0, c > 0 : ∀t, t + ∆t ∈ [a, b], E(|ξt+∆t − ξt |α ) ≤ c|∆t|1+

t❤× q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥

❈❤ø♥❣ ♠✐♥❤✳

{ξt }t∈T

❝ã ➤➵✐ ❞✐Ö♥ ❧✐➟♥ tô❝✳

➜Ó ❣✐➯✐ q✉②Õt ❜➭✐ ♥➭② t❛ ➳♣ ❞ô♥❣ ➤Þ♥❤ ❧ý ♣❤➬♥ ✷✳✷✳✷ tr❛♥❣ ✻✷ ✭❈➳❝ ♠➠ ❤×♥❤ ①➳❝

s✉✃t ✈➭ ø♥❣ ❞ô♥❣✱ ♣❤➬♥ ■■■✱ ◆❣✉②Ô♥ ❉✉② ❚✐Õ♥✮✳

➳♣ ❞ô♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ▼❛r❦♦✈✱ t❛ ❝ã✿

P {|ξt+∆t − ξt | ≥ d} ≤
▲✃②

g(t) = |t|β , 0 < β <

❚❛ t❤✃②

g(t)

❦❤✐

t

c|∆t|1+
E(|ξt+∆t − ξt |α )
≤
dα
dα

(1)

α

✈➭

∞

∞
−n


g(2
n=1

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

)=
n=1

1
2β

n

< ∞ (✈× ✈í✐ β > 0

t❤×

1
< 1)
2β


✷✶

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

❚❤❛②

d = g(∆t) = |∆t|β


✈➭♦ ✭✶✮ t❛ ➤➢î❝✿

P {|ξt+∆t − ξt | ≥ |∆t|β } ≤
▲✃②

q(t) = c.|t|1+ −αβ ✳ ❚❛ t❤✃② q(t)
∞

∞
n

−n

2 q(2

)=

n=1

c
n=1

❦❤✐

1
2

−αβ

❱❐② t❤❡♦ ➤Þ♥❤ ❧ý ♣❤➬♥ ✷✳✷✳✷ tr❛♥❣ ✻✷ t❤×


✶✳✶✹✳

◆Õ✉

{ξt}t∈T

t

c|∆t|1+
= c.|∆t|1+ −αβ
|∆t|α β

) ✈➭
1

n

< ∞ (✈×
{ξt }t∈T

− αβ > 0 ⇒

2

−αβ

< 1)

❝ã ➤➵✐ ❞✐Ö♥ ❧✐➟♥ tô❝✳


❧➭ q✉➳ tr×♥❤ ❣✐❛ sè ➤é❝ ❧❐♣ t❤×

{ξt}t∈T

❧➭ q✉➳

tr×♥❤ ▼❛r❦♦✈
✶✳✶✺✳

❈❤ø♥❣ ♠✐♥❤ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈

❚r♦♥❣ ♣❤➬♥ ♥➭② ♥❤÷♥❣ ❜æ ➤Ò ♠➭ t❤➬② ➤➲ ❝❤ø♥❣ ♠✐♥❤ râ tr♦♥❣ ❣✐➳♦ tr×♥❤ t❤×
❣❤✐ ❧➭ ✧❘å✐✧✳

❇æ ➤Ò ✶✳✶✿

❘å✐

❇æ ➤Ò ✶✳✷✿

❘å✐

❇æ ➤Ò ✶✳✸✿

◆Õ✉

τ1 , τ2

❧➭ ❝➳❝ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈ t❤× ❝➳❝ t❤ê✐ ➤✐Ó♠


τ1 ∧ τ2 := min{τ1 , τ2 }, τ1 ∨ τ2 := max{τ1 , τ2 }, τ1 + τ2
❝ò♥❣ ❧➭ ❝➳❝ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈✳

❈❤ø♥❣ ♠✐♥❤✳

• τ1 ∨ τ2

• τ 1 ∧ τ2

❧➭ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈✿ ❘å✐

❧➭ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈✱ t❤❐t ✈❐②✿

∀t ∈ T : {τ1 ∨ τ2 ≤ t} = {τ1 ≤ t} ∩ {τ2 ≤ t} ∈ Ft
❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻


✷✷

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

• τ1 + τ2

❧➭ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈✱ t❤❐t ✈❐②✿

∀t ∈ T : {τ1 + τ2 ≤ t} = {τ1 = 0, τ2 = t} ∪ {τ1 = t, τ2 = 0}
∪

({τ1 < a} ∩ {τ2 < b}) ∈ Ft

a+b≤t; a,b∈Q; a,b>0

❇æ ➤Ò ✶✳✹✿

❛✮ ◆Õ✉

{τn }n≥1

❧➭ ❞➲② ❝➳❝ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈ t❤×

sup τn

❝ò♥❣ ❧➭ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈✳

n≥1
❜✮ ◆Õ✉ t❤➟♠ ❣✐➯ t❤✐Õt ❞ß♥❣

{Ft } ❧✐➟♥ tô❝ ♣❤➯✐ t❤× inf τn , lim τn , lim τn ❝ò♥❣ ❧➭ ❝➳❝ t❤ê✐ ➤✐Ó♠
n≥1

▼❛r❦♦✈✳

❈❤ø♥❣ ♠✐♥❤✳

❛✮

∀t ∈ T ✱ t❛ ❝ã✿

{sup τn ≤ t} =
n≥1


❜✮

{τn ≤ t} ∈ Ft
n≥1

•{inf τn < t} =
n≥1

{τn < t} ∈ Ft
n≥1

•{lim τn < t} = {inf sup τk < t} ∈ Ft
n≥1 k≥n

•{lim τn < t} = {sup inf τk < t} ∈ Ft
n≥1 k≥n

❇æ ➤Ò ✶✳✺✿

❘å✐

❇æ ➤Ò ✶✳✻✿

❘å✐

❇æ ➤Ò ✶✳✼✿

◆Õ✉


τ, σ

❈❤ø♥❣ ♠✐♥❤✳

❧➭ ❝➳❝ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈ t❤×

{τ < σ}, {τ ≤ σ}, {τ = σ} t❤✉é❝ Fτ ∩ Fσ ✳

• {τ < σ} ∈ Fτ ∩ Fσ ✱ t❤❐t ✈❐②✿ ∀t ∈ T ✱ t❛ ❝ã✿

{τ < σ} ∩ {σ ≤ t} =

({τ < r} ∩ {r < σ ≤ t}) ∈ Ft
r
⇒ {τ < σ} ∈ Fσ

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻


✷✸

❈❤➢➡♥❣✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ▲❚❳❙ ✈➭ ◗❚◆◆

❚➢➡♥❣ tù✿

{τ < σ} ∩ {τ ≤ t} =

({τ ≤ r} ∩ {r < σ} ∩ {τ ≤ t} ∪ {t < σ}) ∈ Ft
r

⇒ {τ < σ} ∈ Fτ ✳ ❱❐② {τ < σ} ∈ Fτ ∩ Fσ
• {τ ≤ σ} = Ω \ {τ > σ} ∈ Fτ ∩ Fσ
• {τ = σ} = {τ ≤ σ} \ {τ < σ} ∈ Fτ ∩ Fσ
✶✳✶✻✳

❱Ý ❞ô ✈Ò t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈

❱Ý ❞ô ✶✳ ●✐➯ sö

C

❧➭ t❐♣ ♠ë tr♦♥❣

❈❤ø♥❣ ♠✐♥❤✳

{(ξt , Ft )}t∈T

{ξt } ❧✐➟♥ tô❝ ♣❤➯✐✱

R✳ ❑❤✐ ➤ã τC := inf{t ≥ 0 : ξt ∈ C} ❧➭ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈✳

rå✐

❱Ý ❞ô ✷✳ ●✐➯ sö

❑❤✐ ➤ã

❧➭ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ ❧✐➟♥ tô❝ ♣❤➯✐✱ ❞ß♥❣

{(ξt , Ft )}t∈T

❧➭ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ ❧✐➟♥ tô❝✱

τD := inf{t ≥ 0 : ξt ∈ D}

❧➭ t❤ê✐ ➤✐Ó♠ ▼❛r❦♦✈ ➤è✐ ✈í✐

D

❧➭ t❐♣ ➤ã♥❣ tr♦♥❣

F := (Ftξ )t∈T

Ftξ := σ(ξs , s ≤ t)✳
❈❤ø♥❣ ♠✐♥❤✳

➜➷t

C := R \ D✳ ❱× C

♠ë ♥➟♥

∃{Kn }n∈N

➤ã♥❣✿

C=

Kn ✳

n∈N

∀s ∈ T ✱ t❛ ❝ã✿
{τD ≤ s} =

n

t
❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

∈
/ Kn } =

n

r
∈
/ Kn } ∈ Fsξ

R✳

✱ tr♦♥❣ ➤ã


❈❤➢➡♥❣ ✷
▼❛rt✐♥❣❛❧❡ tr➟♥ ❦❤♦➯♥❣ t❤ê✐ ❣✐❛♥ rê✐ r➵❝
✈➭ ❧✐➟♥ tô❝

✷✳✶✳

▼ét sè ✈Ý ❞ô ✈Ò ▼❛rt✐♥❣❛❧❡
❱Ý ❞ô ✶✳ ◆Õ✉

▼❛rt✐♥❣❛❧❡ t❤×

X := {(xn , Fn )}1≤n≤N , Y := {(yn , Fn )}1≤n≤N

U := {(un := xn ∧yn , Fn )}1≤n≤N

xn ∨ yn , Fn )}1≤n≤N
❈❤ø♥❣ ♠✐♥❤✳

•

●✐➯ sö

•

X, Y

❧➭ ❝➳❝ s✉♣❡r ✭❤♦➷❝ s✉❜✮

❧➭ s✉♣❡r ▼❛rt✐♥❣❛❧❡ ✭❤♦➷❝

V := {(vn :=

❧➭ s✉❜ ▼❛rt✐♥❣❛❧❡ t➢➡♥❣ ø♥❣✮

◆Õ✉

X, Y

❧➭ s✉♣❡r ▼❛rt✐♥❣❛❧❡ t❤×

❧➭ s✉❜ ▼❛rt✐♥❣❛❧❡✳ ❑❤✐ ➤ã✿

U

❧➭ s✉♣❡r ▼❛rt✐♥❣❛❧❡✿ rå✐

∀n ≥ m✱ t❛ ❝ã✿

E(vn |Fm ) ≥ E(xn |Fm ) ≥ xm
E(vn |Fm ) ≥ E(yn |Fm ) ≥ ym
⇒ E(vn |Fm ) ≥ xm ∨ ym = vm
❱Ý ❞ô ✷✳

∀z ∈ L t❤× X := {(xn , Fn )}1≤n≤N ✱ ✈í✐ xn = E(z|Fn ) ❧➭ ▼❛rt✐♥❣❛❧❡✳

❈❤ø♥❣ ♠✐♥❤✳

∀n ≥ m✱ t❛ ❝ã✿

E(xn |Fm ) = E(E(z|Fn )|Fm ) =(∗) E(z|Fm ) = xm
❱Ý ❞ô ✸✳ ❘å✐

✷✹

✭(∗) ✈×

Fm ⊂ Fn ✮


✷✺

❈❤➢➡♥❣✷✳ ▼❛rt✐♥❣❛❧❡ tr➟♥ ❦❤♦➯♥❣ t❤ê✐ ❣✐❛♥ rê✐ r➵❝ ✈➭ ❧✐➟♥ tô❝

{ηn ∈ L}n≥1 , Eηn = 1, pn :=

❱Ý ❞ô ✹✳ ❈❤♦ ❞➲② ❝➳❝ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣

n

ηi , Fn := σ(η1 , . . . , ηn )✳ ❑❤♦ ➤ã X := {(pn , Fn )}1≤n≤N

❧➭ ▼❛rt✐♥❣❛❧❡✳

i=1

∀n ≥ m✱ t❛ ❝ã✿

❈❤ø♥❣ ♠✐♥❤✳

n

n

E(pn |Fm ) = E(pm .

ηi |Fm ) = pm .E(
i=m+1
n

= pm .E(

ηi |Fm )

i=m+1
n

Eηi = pm , ∀n ≥ 1

ηi ) = p m .

i=m+1

i=m+1

❱Ý ❞ô ✺✳ ❈❤♦ ❞➲② ❝➳❝ ➤➵✐ ❧➢î♥❣ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➞♠

{ηn ∈ L}n≥1 , Sn =

n

ηi , Fn :=
i=1

σ(η1 , . . . , ηn )✳ ❑❤✐ ➤ã X := {(Sn , Fn )}1≤n≤N

❈❤ø♥❣ ♠✐♥❤✳

❧➭ s✉❜ ▼❛rt✐♥❣❛❧❡✳

∀n ≥ m✱ t❛ ❝ã✿
m

E(Sn |Fm ) = E[(

n

ηi +
i=1

m

=

ηi )|Fm ]
i=m+1
n

ηi + E(
i=1

ηi )|Fm )

i=m+1

≥ Sm

X := {(xn , Fn )}1≤n≤N

❱Ý ❞ô ✻✳ ◆Õ✉

∞, ∀n t❤× Y := {(f (xn ), Fn )}1≤n≤N
❈❤ø♥❣ ♠✐♥❤✳

❧➭ ▼❛rt✐♥❣❛❧❡✱ ❤➭♠

f : R → R ❧å✐ ✈í✐ E|f (xn )| <

❧➭ s✉❜ ▼❛rt✐♥❣❛❧❡✳

∀n ≥ m✱ ➳♣ ❞ô♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❏❡♥s❡♥✱ t❛ ❝ã✿
E(f (xn )|Fm ) ≥ f (E(xn |Fm )) = f (xm )

❱Ý ❞ô ✼✳ ◆Õ✉

❧å✐ ✈í✐
♥Õ✉

X := {(xn , Fn )}1≤n≤N

E|f (xn )| < ∞, ∀n

X := {(xn , Fn )}1≤n≤N

❚r➢➡♥❣ ◆❣ä❝ ❍➯✐ ✲ ❑✶✻

t❤×

❧➭ s✉❜ ▼❛rt✐♥❣❛❧❡✱ ❤➭♠

Y := {(f (xn ), Fn )}1≤n≤N

❧➭ ▼❛rt✐♥❣❛❧❡ t❤×

f :R→R

❦❤➠♥❣ ❣✐➯♠✱

❧➭ s✉❜ ▼❛rt✐♥❣❛❧❡✳ ➜➷❝ ❜✐Öt✱

{E|xn |p }1≤n≤N

❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠