Martingale and Almost sure Convergence

Durrett 책 일부를 정리한 내용입니다.

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$H_n$ is predictable sequence if $H_n \in \mathcal{F}_{n-1}$.

Theorem 4.2.8 Let $X_n$ be a supermartingale. If $H_n \geq 0$ is predictable and each $H_n$ is bounded then $(H\cdot X)_n$ is a supermartingale.

Proof

\begin{align} E ( (H\cdot X)_{n+1} | \mathcal{F}_n) &= (H \cdot X)_n + E (H_{n+1} (X_{n+1}-X_n) | \mathcal{F}_n)\\ &= (H \cdot X)_n + H_{n+1} E((X_{n+1}-X_n)|\mathcal{F}_n) \leq (H\cdot X)_n\end{align} 

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Stopping time

A random variable $N$ is said to be stopping time if $\{N=n\} \in \mathcal{F}_n$ for all $n<\infty$. 

Let $H_n = 1_{\{N\geq n\}}$ then $\{N\geq n\} = \{ N \leq n-1 \}^c \in \mathcal{F}_{n-1}$ 

$X_{n \wedge N} - X_0 = (H \cdot X)_n$ and this is supermartingale if $X_n$ is supermartingale by Theorem 4.2.8.

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Upcrossing

$N_{2k} = \inf \{ m> N_{2k-2} : X_m \leq a \}$

$N_{2k-1} = \inf \{ m> N_{2k-1}  : X_m \geq b  \}$

$H_n =$

\begin{cases} 1 & \text{if} \qquad N_{2k-1} < m \leq N_{2k} \qquad \text{for some } k\\0 &\text{otherwise}\end{cases}

Theorem 4.2.10 ( Upcrossing Inequality ) If $X_m$ is a submartingale, then 

$(b-a) EU_n \leq E(X_n-a)^+ - E(X_0-a)^+$

Proof)

$(b-a) U_n \leq (H\cdot Y)_n$

$Y_n = a + (X_n-a)^+$ 이라고 할 때, 

$Y_n -Y_0 = (H \cdot Y)_n + ((1-H)\cdot Y)_n$