假設 $X_n, n\ge 0 $ 為 submartingale。令 $a<b$ 且 $N_0 := -1$ 與 對任意 $k \ge 1$ 我們定義 stopping time 如下: \[\begin{array}{l} {N_{2k - 1}}: = \inf \left\{ {m > {N_{2k - 2}}:{X_m} \le a} \right\}\\ {N_{2k}}: = \inf \left\{ {m > {N_{2k - 1}}:{X_m} \ge b} \right\} \end{array} \] 且我們觀察以下事件 \[\begin{array}{l} \{ {N_{2k - 1}} < m \le {N_{2k}}\} = \{ {N_{2k - 1}} < m\} \cap \{ m \le {N_{2k}}\} \\ \begin{array}{*{20}{c}} {}&{}&{}&{}&{}&{}&{}&{}&{} \end{array} = \underbrace {\{ {N_{2k - 1}} \le m - 1\} }_{ \in {F_{m - 1}}} \cap \underbrace {{{\{ {N_{2k}} \le m - 1\} }^c}}_{ \in {F_{m - 1}}} \in {F_{m - 1}} \end{array}\] ============== Fact: 若 $X_m, m\ge 0$ 為 submartingale 則 \[ (b-a)EU_n \le E(X_n -a)^+ - E(X_0 - a)^+ \]其中 ${U_n}: = \sup \left\{ {k:{N_{2k}} \le n} \right\}$ 表示在 時間$n$ 之前 往上穿越的個數 ============== Proof: omitted. ========== Theorem: Martingale Convergence Theorem 若 $X_n$ 為 submartingale 滿足 $\sup E[X_n^+] < \infty$ 則存在 $X
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