謝宗翰的隨筆

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