정지 시간의 성질들
📂확률론 정지 시간의 성질들 정리 확률 공간 ( Ω , F , P ) ( \Omega , \mathcal{F} , P) ( Ω , F , P ) 마틴 게일 { ( X n , F n ) } \left\{ ( X_{n} , \mathcal{F}_{n} ) \right\} { ( X n , F n ) } 정지 시간 τ \tau τ F τ : = { A ∈ F : A ∩ ( τ = n ) ∈ F n } \mathcal{F}_{\tau}:= \left\{ A \in \mathcal{F}: A \cap ( \tau = n ) \in \mathcal{F}_{n} \right\} F τ := { A ∈ F : A ∩ ( τ = n ) ∈ F n } τ \tau τ
[1]: F τ \mathcal{F}_{\tau} F τ [2]: τ \tau τ F τ \mathcal{F}_{\tau} F τ [3]: 마틴 게일 { ( X n , F n ) } \left\{ ( X_{n} , \mathcal{F}_{n} ) \right\} { ( X n , F n ) } X τ X_{\tau} X τ F τ \mathcal{F}_{\tau} F τ [4]: 1 ( σ = n ) \mathbb{1}_{(\sigma = n)} 1 ( σ = n ) F σ \mathcal{F}_{\sigma} F σ [5]: Z n Z_{n} Z n F n F_{n} F n Z n 1 σ = n Z_{n} \mathbb{1}_{\sigma = n} Z n 1 σ = n F σ \mathcal{F}_{\sigma} F σ F n \mathcal{F}_{n} F n Z n 1 ( σ = n ) = Z σ 1 ( σ = n ) Z_{n} \mathbb{1}_{(\sigma = n)} = Z_{\sigma} \mathbb{1}_{(\sigma = n)} Z n 1 ( σ = n ) = Z σ 1 ( σ = n ) [6]: E ( X τ ∣ F σ ) 1 ( σ = n ) = E ( X τ ∣ F n ) 1 ( σ = n ) a.s. E(X_{\tau} | \mathcal{F}_{\sigma} ) \mathbb{1}_{(\sigma = n)} = E(X_{\tau} | \mathcal{F}_{n} ) \mathbb{1}_{(\sigma = n)} \text{ a.s.} E ( X τ ∣ F σ ) 1 ( σ = n ) = E ( X τ ∣ F n ) 1 ( σ = n ) a.s. 보렐 셋 B ∈ B ( R ) B \in \mathcal{B}(\mathbb{R}) B ∈ B ( R ) ( τ ∈ B ) = τ − 1 ( B ) (\tau \in B) = \tau^{-1} (B) ( τ ∈ B ) = τ − 1 ( B ) ( τ = n ) (\tau = n) ( τ = n ) τ − 1 ( { n } ) \tau^{-1} ( \left\{ n \right\} ) τ − 1 ( { n } ) τ \tau τ F n \mathcal{F}_{n} F n 가측 함수 라는 것은 모든 보렐 셋 B ∈ B ( R ) B \in \mathcal{B}(\mathbb{R}) B ∈ B ( R ) τ − 1 ( B ) ∈ F n \tau^{-1} (B) \in \mathcal{F}_{n} τ − 1 ( B ) ∈ F n 확률 과정 { X n } n ∈ N 0 \left\{ X_{n} \right\}_{n \in \mathbb{N}_{0}} { X n } n ∈ N 0 ω ∈ Ω \omega \in \Omega ω ∈ Ω X τ X_{\tau} X τ X τ = X τ ( ω ) = X τ ( ω ) ( ω )
X_{\tau} = X_{\tau} ( \omega )= X_{\tau (\omega)} ( \omega )
X τ = X τ ( ω ) = X τ ( ω ) ( ω ) 설명 정지 시간 은 본질적으로 우리가 원하는 어떤 타이밍을 나타내기 위한 확률 변수로 이해할 수 있다.
증명 [1] F τ \mathcal{F}_{\tau} F τ 시그마 필드 의 조건을 만족하는지 확인하면 된다.
Part (i). ∅ ∈ F τ \emptyset \in \mathcal{F}_{\tau} ∅ ∈ F τ
F \mathcal{F} F F n \mathcal{F}_{n} F n ∅ \emptyset ∅ ∅ ∈ F τ \emptyset \in \mathcal{F}_{\tau} ∅ ∈ F τ
Part (ii). A ∈ F τ ⟹ A c ∈ F τ A \in \mathcal{F}_{\tau} \implies A^{c} \in \mathcal{F}_{\tau} A ∈ F τ ⟹ A c ∈ F τ
A c ∩ ( τ = n ) = ( τ = n ) ∖ [ A ∩ ( τ = n ) ]
A^{c} \cap ( \tau = n ) = ( \tau = n ) \setminus \left[ A \cap ( \tau = n ) \right]
A c ∩ ( τ = n ) = ( τ = n ) ∖ [ A ∩ ( τ = n ) ] ( τ = n ) ∈ F n ( \tau = n) \in \mathcal{F}_{n} ( τ = n ) ∈ F n A ∈ F τ A \in \mathcal{F}_{\tau} A ∈ F τ F τ F_{\tau} F τ A ∩ ( τ = n ) ∈ F n A \cap ( \tau = n) \in \mathcal{F}_{n} A ∩ ( τ = n ) ∈ F n A c ∈ F τ A^{c} \in \mathcal{F}_{\tau} A c ∈ F τ
Part (iii). { A n } n = 1 ∞ ⊂ F τ ⟹ ⋃ i = 1 ∞ A i ∈ F τ \displaystyle \left\{ A_{n} \right\}_{n=1}^{\infty} \subset \mathcal{F}_{\tau} \implies \bigcup_{i=1}^{\infty} A_{i} \in \mathcal{F}_{\tau} { A n } n = 1 ∞ ⊂ F τ ⟹ i = 1 ⋃ ∞ A i ∈ F τ
i = 1 , 2 , ⋯ i = 1 , 2, \cdots i = 1 , 2 , ⋯ A i ∈ F τ A_{i} \in \mathcal{F}_{\tau} A i ∈ F τ F τ F_{\tau} F τ A i ∩ ( τ = n ) ∈ F n A_{i} \cap ( \tau = n) \in \mathcal{F}_{n} A i ∩ ( τ = n ) ∈ F n ⋃ i = 1 ∞ A i ∩ ( τ = n ) = ⋃ i = 1 ∞ [ A i ∩ ( τ = n ) ] ∈ F n
\bigcup_{i=1}^{\infty} A_{i} \cap (\tau = n ) = \bigcup_{i=1}^{\infty} \left[ A_{i} \cap ( \tau = n) \right] \in \mathcal{F}_{n}
i = 1 ⋃ ∞ A i ∩ ( τ = n ) = i = 1 ⋃ ∞ [ A i ∩ ( τ = n ) ] ∈ F n F τ F_{\tau} F τ ⋃ i = 1 ∞ A i ∈ F τ \bigcup_{i=1}^{\infty} A_{i} \in \mathcal{F}_{\tau} i = 1 ⋃ ∞ A i ∈ F τ
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[2] 가측 함수의 동치조건 에 따라 τ \tau τ F τ \mathcal{F}_{\tau} F τ k ∈ R k \in \mathbb{R} k ∈ R ( τ ≤ k ) ∈ F τ ( \tau \le k ) \in \mathcal{F}_{\tau} ( τ ≤ k ) ∈ F τ ( τ ≤ k ) ∩ ( τ = n ) = { ∅ , k < n ( τ = n ) , k ≥ n
( \tau \le k ) \cap ( \tau = n) = \begin{cases} \emptyset &, k < n
\\ (\tau = n) &, k \ge n \end{cases}
( τ ≤ k ) ∩ ( τ = n ) = { ∅ ( τ = n ) , k < n , k ≥ n ∅ ∈ F n \emptyset \in \mathcal{F}_{n} ∅ ∈ F n ( τ = n ) ∈ F n (\tau =n ) \in \mathcal{F}_{n} ( τ = n ) ∈ F n k ∈ R k \in \mathbb{R} k ∈ R ( τ ≤ k ) ∩ ( τ = n ) ∈ F n ( \tau \le k ) \cap ( \tau = n) \in \mathcal{F}_{n} ( τ ≤ k ) ∩ ( τ = n ) ∈ F n k ∈ R k \in \mathbb{R} k ∈ R ( τ ≤ k ) ∈ F τ ( \tau \le k ) \in \mathcal{F}_{\tau} ( τ ≤ k ) ∈ F τ τ \tau τ F τ \mathcal{F}_{\tau} F τ
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[3] 임의의 보렐 셋 B ∈ B ( R ) B \in \mathcal{B}(\mathbb{R}) B ∈ B ( R ) ( X τ ∈ B ) ∩ ( τ = n ) = ( X n ∈ B ) ∩ ( τ = n )
(X_{\tau} \in B) \cap (\tau = n) = (X_{n} \in B) \cap (\tau = n)
( X τ ∈ B ) ∩ ( τ = n ) = ( X n ∈ B ) ∩ ( τ = n ) ( X τ ∈ B ) (X_{\tau} \in B) ( X τ ∈ B ) ( X n ∈ B ) (X_{n} \in B) ( X n ∈ B ) ( τ = n ) (\tau = n) ( τ = n ) 교집합 을 취했을 때 오직 τ = n \tau = n τ = n 조건부 기대값 E ( Y ∣ X ) E(Y|X) E ( Y ∣ X ) X = x X=x X = x 마틴게일 의 정의에 따라 ( X n ∈ B ) ∈ F n (X_{n} \in B) \in \mathcal{F}_{n} ( X n ∈ B ) ∈ F n 정지 시간 의 정의에 따라 ( τ = n ) ∈ F n (\tau = n ) \in \mathcal{F}_{n} ( τ = n ) ∈ F n [ ( X τ ∈ B ) ∩ ( τ = n ) ] ∈ F τ \left[ (X_{\tau} \in B) \cap (\tau = n) \right] \in \mathcal{F}_{\tau} [ ( X τ ∈ B ) ∩ ( τ = n ) ] ∈ F τ X τ X_{\tau} X τ F τ \mathcal{F}_{\tau} F τ
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[4] 사건 A ∈ F A \in \mathcal{F} A ∈ F 1 A \mathbb{1}_{A} 1 A ( 1 A ≤ a ) = { Ω , a ≥ 1 A c , a ∈ [ 0 , 1 ) ∅ , a < 0
( \mathbb{1}_{A} \le a ) = \begin{cases} \Omega &, a \ge 1
\\ A^{c} &, a \in [0,1)
\\ \emptyset &, a < 0 \end{cases}
( 1 A ≤ a ) = ⎩ ⎨ ⎧ Ω A c ∅ , a ≥ 1 , a ∈ [ 0 , 1 ) , a < 0 A = ( σ = n ) A = (\sigma = n) A = ( σ = n ) 보렐 셋 B ∈ B ( R ) B \in \mathcal{B}(\mathbb{R}) B ∈ B ( R ) ∅ , ( σ ≠ n ) , Ω ∈ F σ \emptyset, (\sigma \ne n), \Omega \in \mathcal{F}_{\sigma} ∅ , ( σ = n ) , Ω ∈ F σ ( 1 ( σ = n ) ≤ a ) ∈ F σ ( \mathbb{1}_{(\sigma = n)} \le a ) \in \mathcal{F}_{\sigma} ( 1 ( σ = n ) ≤ a ) ∈ F σ 1 ( σ = n ) \mathbb{1}_{(\sigma = n)} 1 ( σ = n ) F σ \mathcal{F}_{\sigma} F σ
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[5] Z n 1 ( σ = n ) = { Z σ ⋅ 1 , σ = n 0 , σ ≠ 0
Z_{n} \mathbb{1}_{(\sigma = n)} = \begin{cases} Z_{\sigma} \cdot 1 &, \sigma = n
\\ 0 &, \sigma \ne 0 \end{cases}
Z n 1 ( σ = n ) = { Z σ ⋅ 1 0 , σ = n , σ = 0 Z n 1 ( σ = n ) = Z σ 1 ( σ = n ) Z_{n} \mathbb{1}_{(\sigma = n)} = Z_{\sigma} \mathbb{1}_{(\sigma = n)} Z n 1 ( σ = n ) = Z σ 1 ( σ = n ) Z σ Z_{\sigma} Z σ F σ \mathcal{F}_{\sigma} F σ 1 ( σ = n ) \mathbb{1}_{(\sigma = n)} 1 ( σ = n ) F σ \mathcal{F}_{\sigma} F σ Z n 1 ( σ = n ) Z_{n} \mathbb{1}_{(\sigma = n)} Z n 1 ( σ = n ) F σ \mathcal{F}_{\sigma} F σ
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[6] E ( X τ ∣ F n ) E \left( X_{\tau} | \mathcal{F}_{n} \right) E ( X τ ∣ F n ) F n \mathcal{F}_{n} F n E ( X τ ∣ F n ) 1 σ = n E \left( X_{\tau} | \mathcal{F}_{n} \right) \mathbb{1}_{ \sigma = n} E ( X τ ∣ F n ) 1 σ = n F σ \mathcal{F}_{\sigma} F σ 1 ( σ = n ) \mathbb{1}_{(\sigma=n)} 1 ( σ = n ) 스무딩 성질 에 의해 F σ \mathcal{F}_{\sigma} F σ E ( ⋅ ∣ F σ ) E ( \cdot | \mathcal{F}_{\sigma} ) E ( ⋅ ∣ F σ )
스무딩 성질 : X X X G \mathcal{G} G E ( X Y ∣ G ) = X E ( Y ∣ G ) a.s. E(XY | \mathcal{G}) = X E (Y | \mathcal{G}) \text{ a.s.} E ( X Y ∣ G ) = XE ( Y ∣ G ) a.s.
조건부 기대값의 성질 : X X X F \mathcal{F} F E ( X ∣ F ) = X a.s. E(X|\mathcal{F}) =X \text{ a.s.} E ( X ∣ F ) = X a.s.
한편 위와 같은 성질들에 따라, 모든 A ∈ F σ A \in \mathcal{F}_{\sigma} A ∈ F σ ∫ A E ( X τ ∣ F σ ) 1 ( σ = n ) d P = ∫ A E ( X τ 1 ( σ = n ) ∣ F σ ) d P = ∫ A X τ 1 ( σ = n ) d P = ∫ A ∩ ( σ = n ) X τ d P = ∫ A X τ 1 ( σ = n ) d P = ∫ A ∩ ( σ = n ) E ( X τ ∣ F n ) d P = ∫ A E ( X τ ∣ F n ) 1 ( σ = n ) d P
\begin{align*}
\int_{A} E \left( X_{\tau} | \mathcal{F}_{\sigma} \right) \mathbb{1}_{(\sigma = n)} dP =& \int_{A} E \left( X_{\tau} \mathbb{1}_{(\sigma = n)} | \mathcal{F}_{\sigma} \right) dP
\\ =& \int_{A} X_{\tau} \mathbb{1}_{(\sigma = n)} dP
\\ =& \int_{A \cap (\sigma = n)} X_{\tau} dP
\\ =& \int_{A} X_{\tau} \mathbb{1}_{(\sigma = n)} dP
\\ =& \int_{A \cap (\sigma = n)} E \left( X_{\tau} | \mathcal{F}_{n} \right) dP
\\ =& \int_{A } E \left( X_{\tau} | \mathcal{F}_{n} \right) \mathbb{1}_{(\sigma = n)} dP
\end{align*}
∫ A E ( X τ ∣ F σ ) 1 ( σ = n ) d P = = = = = = ∫ A E ( X τ 1 ( σ = n ) ∣ F σ ) d P ∫ A X τ 1 ( σ = n ) d P ∫ A ∩ ( σ = n ) X τ d P ∫ A X τ 1 ( σ = n ) d P ∫ A ∩ ( σ = n ) E ( X τ ∣ F n ) d P ∫ A E ( X τ ∣ F n ) 1 ( σ = n ) d P ∀ A ∈ F , ∫ A f d m = 0 ⟺ f = 0 a.e. \displaystyle \forall A \in \mathcal{F}, \int_{A} f dm = 0 \iff f = 0 \text{ a.e.} ∀ A ∈ F , ∫ A fd m = 0 ⟺ f = 0 a.e. E ( X τ ∣ F σ ) 1 ( σ = n ) = E ( X τ ∣ F n ) 1 ( σ = n ) a.s.
E(X_{\tau} | \mathcal{F}_{\sigma} ) \mathbb{1}_{(\sigma = n)} = E(X_{\tau} | \mathcal{F}_{n} ) \mathbb{1}_{(\sigma = n)} \text{ a.s.}
E ( X τ ∣ F σ ) 1 ( σ = n ) = E ( X τ ∣ F n ) 1 ( σ = n ) a.s.
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