[機率論] Martingale (3) - Example

Example: 令 $\{X_n\}$ 為 Martingale with Filtration $\mathcal{F_n}$，假設 $T$ 為 stopping time。試證 $Y_n:=X_{\min(n,T)}$ 為 Martingale with $\mathcal{F_n}$。

Proof:
首先證明 (1) $E|Y_n| < \infty $：
注意到：
\[\begin{array}{l}
{Y_n}: = {X_{n \wedge T}} = {X_n}{1_{n{\rm{ < }}T}} + {X_T}{1_{T \le n}}\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} = {X_n}{1_{n{\rm{ < }}T}} + \sum\limits_{m = 1}^n {{X_T}{1_{T = k}}}
\end{array}\]故
\[\begin{array}{l}
E\left| {{Y_n}} \right| = E\left| {{X_{n \wedge T}}} \right| = E\left| {{X_n}{1_{n{\rm{ < }}T}} + \sum\limits_{m = 1}^n {{X_T}{1_{T = k}}} } \right|\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} \le E\left| {{X_n}{1_{n{\rm{ < }}T}}} \right| + \sum\limits_{m = 1}^n {E\left| {{X_T}{1_{T = k}}} \right|} \\
\begin{array}{*{20}{c}}
{}&{}&{}&{}
\end{array} \le E\left| {{X_n}} \right| + \sum\limits_{m = 1}^n {E\left| {{X_T}} \right|}  < \infty
\end{array}\]

接著我們證明 (2) : $Y_n \in \mathcal{F}_n $
觀察 \[{Y_n}: = {X_{n \wedge T}} = {X_n}{1_{n{\rm{ < }}T}} + \sum\limits_{m = 1}^n {{X_T}{1_{T = k}}} \]由於 $1_{T>n} = 1_{T \le n}^c \in \mathcal{F}_n$ 且 $X_m \in \mathcal{F}_m \subset \mathcal{F}_n$ 且 $1_{T=m} \in \mathcal{F}_m \subset \mathcal{F}_n$ 對任意 $m\le n$ 故
\[{Y_n}: = \underbrace {{X_n}}_{ \in {F_n}}\underbrace {{1_{n{\rm{ < }}T}}}_{ \in {F_n}} + \underbrace {\sum\limits_{m = 1}^n {{X_T}{1_{T = k}}} }_{ \in {F_n}} \in {F_n}\]

最後我們證明 (3): $E[Y_{n+1}|\mathcal{F}_n] = E[X_{}]$ 注意到
\[\begin{array}{l}
E[{Y_{n + 1}}|{{{\cal F}}_n}] = E[{X_{n + 1}}{1_{n + 1 < T}} + {X_T}{1_{T \le n + 1}}|{{{\cal F}}_n}]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}&{}&{}
\end{array} = E[{X_{n + 1}}{1_{n + 1 < T}}|{{{\cal F}}_n}] + E[{X_T}{1_{T \le n + 1}}|{{{\cal F}}_n}]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}&{}&{}
\end{array} = E[{X_{n + 1}}{1_{n + 1 < T}}|{{{\cal F}}_n}] + E[\sum\limits_{k = 1}^n {{X_k}{1_{T = k}}}  + {X_{n + 1}}{1_{T = n + 1}}|{{{\cal F}}_n}]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}&{}&{}
\end{array} = E[{X_{n + 1}}{1_{n + 1 < T}}|{{{\cal F}}_n}] + {X_T}{1_{T \le n}} + E[{X_{n + 1}}{1_{T = n + 1}}|{{{\cal F}}_n}]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}&{}&{}
\end{array} = {X_n}{1_{n + 1 \le T}} + {X_T}{1_{T \le n}}\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}&{}&{}
\end{array} = {X_n}{1_{n < T}} + {X_T}{1_{T \le n}} = {Y_n}
\end{array}\]