随机过程 | 鞅论 1.1 Doob下鞅分解定理 & 1.2停时

2026-08-08 18:59:39 星期六
好焦虑，怎么又要开学了。开个新笔记假装督促一下自己(❁´◡`❁)

1.1 下鞅分解定理

\((\Omega, \mathcal{F}, P)\) 为概率空间，指标集 \(T = \mathbb{Z}_+\) 或 \(\mathbb{R}_+\)

Def 1.1.1 \(\{\mathcal{F}_t\}_{t \in T}\)，\(\mathcal{F}_t \subset \mathcal{F}\) 为子 \(\sigma\)-代数，如果 \(\forall s \leq t\) 有 \(\mathcal{F}_s \subset \mathcal{F}_t\)，则 \(\{\mathcal{F}_t\}\) 为流.

Def 1.1.2 \(\{X_t\}_{t \in T}\)，若 \(\forall t \in T\), \(X_t \in \mathcal{F}_t\)，则 \(\{X_t\}_{t \in T}\) 关于 \(\{\mathcal{F}_t\}_{t \in T}\) 适应.

Def 1.1.3 考虑 \(\{X_t, \mathcal{F}_t\}_{t \in T}\)，若满足：

① \(X_t \in \mathcal{F}_t\) \(\quad \forall t \in T\)
② \(E|X_t| < +\infty\)
③ \(\forall s \leq t\) \(E(X_t | \mathcal{F}_s) = X_s\) a.s.

则 \(\{X_t, \mathcal{F}_t\}_{t \in T}\) 为鞅。若 \(E(X_t | \mathcal{F}_s) \leq X_s\)，则为上鞅； \(E(X_t | \mathcal{F}_s) \geq X_s\) 下鞅.

Remark 1.1.4 \(E[E(X_t | \mathcal{F}_s)] = EX_t = EX_s\)，说明期望都相等.

Prop 1.1.5 设 \(\{X_t\}, \{Y_t\}\) 关于 \(\{\mathcal{F}_t\}_{t \in T}\) 是上鞅，则有：

(1) \(EX_t\) 非增
(2) \(\{-X_t, \mathcal{F}_t\}_{t \in T}\) 为下鞅
(3) \(\forall a,b > 0\)，\(\{aX_t + bY_t\}_{t \in T}\) 为上鞅
(4) \(\{X_t \wedge Y_t\}_{t \in T}\) 为上鞅.

Pf. (1) \(E[X_t | \mathcal{F}_s] \leq X_s\) \(\Rightarrow\) \(EX_t \leq EX_s\)

(2) \(\forall s \leq t\), \(E[-X_t | \mathcal{F}_s] = -E[X_t | \mathcal{F}_s] \geq -X_s\)

(3) \(\forall s \leq t\), \(E[aX_t + bY_t | \mathcal{F}_s] = aE[X_t | \mathcal{F}_s] + bE[Y_t | \mathcal{F}_s] \leq aX_s + bY_s\)

(4) \(\forall s \leq t\) \(E[X_t \wedge Y_t | \mathcal{F}_s] \leq E[X_t | \mathcal{F}_s] \leq X_s\)
\(E[X_t \wedge Y_t | \mathcal{F}_s] \leq E[Y_t | \mathcal{F}_s] \leq Y_s\)
\(\Rightarrow\) \(E[X_t \wedge Y_t | \mathcal{F}_s] \leq X_s \wedge Y_s\) a.s.

Prop 1.1.6

  1. 若 \(\{X_t, \mathcal{F}_t\}_{t \in T}\) 为鞅，\(\varphi\) 为凸函数，\(E|\varphi(X_t)| < +\infty\), \(\forall t \in T\)，则 \(\{\varphi(X_t), \mathcal{F}_t\}\) 为下鞅。

  2. \(\{X_t, \mathcal{F}_t\}_{t \in T}\) 为下鞅，\(\varphi\) 为单增凸函数，\(\varphi(X_t) \in L^1(P)\)，\(\{\varphi(X_t), \mathcal{F}_t\}\) 下鞅。

Pf. 1 \(\forall s \leq t\) \(E[\varphi(X_t) | \mathcal{F}_s] \geq \varphi(E[X_t | \mathcal{F}_s]) = \varphi(X_s)\)

而且 \(\varphi(X_t)\) 关于 \(\mathcal{F}_t\) 可测，\(E[\varphi(X_t)] < +\infty\).

  1. \(\varphi(X_t) \in \mathcal{F}_t\), \(E|\varphi(t)| < +\infty\)（因为 \(\varphi(X_t) \in L^1(P)\)）
    \(E[\varphi(X_t) | \mathcal{F}_s] \geq \varphi(E[X_t | \mathcal{F}_s]) \geq \varphi(X_s)\)

Example 1.1.7 1. \(X \in L^1(\Omega, \mathcal{F}, P)\) \(\{\mathcal{F}_t\}_{t \in T}\) 为 \(\sigma\)-代数流，令 \(Y_t = E(X | \mathcal{F}_t)\)，则 \(\forall s < t\)
\(E[Y_t | \mathcal{F}_s] = E[E(X | \mathcal{F}_t) | \mathcal{F}_s] = E(X | \mathcal{F}_s) = Y_s\) a.s.

  1. \(X_n = X_0 + \sum_{i=1}^n Y_i\)，\(Y_i = \begin{cases} 1 & p=1/2 \\ -1 & p=1/2 \end{cases}\)，\(E|X_0| < +\infty\)，\(\mathcal{F}_n = \sigma(X_0, X_1, \cdots, Y_n)\)，\(\forall m \leq n\)，\(Y_i\) 独立同分布

\(E(X_n | \mathcal{F}_m) = E(X_0 + \sum_{i=1}^m Y_i + \sum_{i=m+1}^n Y_i | \mathcal{F}_m) = X_0 + \sum_{i=1}^m Y_i + \sum_{k=m+1}^n E(Y_k | \mathcal{F}_m) = X_0 + \sum_{i=1}^m Y_i = X_m\)

\(\{X_n, \mathcal{F}_n\}_{n=1}^{\infty}\) 为鞅。

Thm 1.1.8（Doob 下鞅分解定理） 设 \(\{X_n, \mathcal{F}_n\}_{n=1}^{\infty}\) 为下鞅，则存在唯一的 \(\{M_n\}_{n=1}^{\infty}\), \(\{A_n\}_{n=1}^{\infty}\) 满足：

  1. \(X_n = M_n + A_n\)
  2. \(\{M_n, \mathcal{F}_n\}_{n \geq 1}\) 是鞅
  3. \(A_1 = 0\)，\(A_n \in \mathcal{F}_{n-1}\)，\(A_n\) 单增（因此 \(A_n \geq 0\)）.

Pf. 我们先设有这样的分解 \(X_n = M_n + A_n\)。则

\[\begin{aligned} X_n - X_{n-1} &= (M_n - M_{n-1}) + (A_n - A_{n-1}) \\ E(X_n - X_{n-1} | \mathcal{F}_{n-1}) &= 0 + A_n - A_{n-1} \\ E(X_n | \mathcal{F}_{n-1}) - X_{n-1} &= A_n - A_{n-1} \\ \Rightarrow A_n &= \sum_{i=2}^n [E(X_i | \mathcal{F}_{i-1}) - X_{i-1}] \end{aligned} \]

定义 \(M_n = X_n - A_n\)，下证 \(\{M_n, \mathcal{F}_n\}_{n=1}^{\infty}\) 为鞅：

  1. \(M_n \in \mathcal{F}_n\)：\(\forall n\)，\(X_n \in \mathcal{F}_n\)，\(A_n \in \mathcal{F}_{n-1} \subset \mathcal{F}_n\) \(\Rightarrow\) \(M_n \in \mathcal{F}_n\)

  2. \(E|M_n| < +\infty\)：\(E|M_n| \leq E|X_n| < +\infty\)

  3. \(E(M_n | \mathcal{F}_{n-1}) = E(X_n | \mathcal{F}_{n-1}) - E(A_n | \mathcal{F}_{n-1}) = X_{n-1} - A_{n-1} = M_{n-1}\)

最后证唯一性：设 \(\{A_n'\}_{n=1}^{\infty}\)，\(\{M_n'\}_{n=1}^{\infty}\) 也满足 \(X_n = M_n' + A_n'\)

\[\begin{aligned} M_n - M_n' &= A_n' - A_n \quad (1)\\ \Rightarrow E(M_n - M_n' | \mathcal{F}_{n-1}) &= E(A_n' - A_n | \mathcal{F}_{n-1}) \\ M_{n-1} - M_{n-1}' &= A_n' - A_n \quad (2) \end{aligned} \]

在（1）中令 \(n=1\)，有 \(M_1 = M_1'\)；在（2）中，令 \(n=2\)，\(\Rightarrow A_2' = A_2\)，以此类推，有唯一性。\(\square\)

Def 1.1.9（一致可积） 称一族随机变量 \(\{X_t\}_{t \in T}\) 一致可积，若

\[\lim_{\lambda \to +\infty} \sup_{t \in T} \int_{\{|X_t| \geq \lambda\}} |X_t| dP = 0 \]

Remark 1.1.10 由 Def 1.1.9 可知，

\[E|X_t| = \int_{|X_t| \geq \lambda} |X_t| dP + \int_{|X_t| \leq \lambda} |X_t| dP \leq \lambda + \int_{|X_t| \geq \lambda} |X_t| dP \leq \lambda + 1 \]

Prop 1.1.11 \(\{X_t\}\) 一致可积 \(\Leftrightarrow\) 成立：

  1. \(\{X_t\}_{t \in T}\) 在 \(L^1(\Omega, \mathcal{F}, P)\) 中有界
  2. \(\forall \varepsilon > 0\)，\(\exists \delta > 0\)，\(\forall A \in \mathcal{F}\)，若 \(P(A) < \delta\)，\(\int_A |X_t| dP < \varepsilon\)

证明

(\(\Rightarrow\)) 1. 已证.

\[\int_A |X_t| dP = \left( \int_{A \cap \{|X_t| \geq \lambda\}} + \int_{A \cap \{|X_t| < \lambda\}} \right) |X_t| dP \leq \lambda P(A) + \int_{\{|X_t| \geq \lambda\}} |X_t| dP. \]

由于一致可积，\(\forall \varepsilon > 0\)，\(\exists \lambda > 1\)，使得

\[\int_{\{|X_t| \geq \lambda\}} |X_t| dP < \frac{\varepsilon}{2}. \]

取 \(\delta = \frac{\varepsilon}{2\lambda}\)，则当 \(P(A) < \delta\) 时，有

\[\int_A |X_t| dP < \varepsilon. \]

(\(\Leftarrow\)) 注意

\[P(|X_t| \geq \lambda) \leq \frac{1}{\lambda} E|X_t| \to 0 \quad (\lambda \to \infty), \]

且 \(E|X_t| \leq C\)，\(\forall t \in T\). 故当 \(\lambda > \frac{C}{\delta}\) 时，有 \(P(|X_t| \geq \lambda) \leq \delta\)，由 2 得

\[\int_{\{|X_t| \geq \lambda\}} |X_t| dP < \varepsilon. \]

Prop 1.1.12 若 \(\{X_t\}_{t \in T}\)，\(\{Y_t\}_{t \in T}\) 都一致可积，则 \(\{X_t \pm Y_t\}_{t \in T}\) 也一致可积.

证明. 借助 Prop 1.1.11 验证：

  1. \(X_t \pm Y_t \in L^1(\Omega, \mathcal{F}, P)\).

  2. \(\forall \varepsilon > 0\)，\(\exists \delta > 0\)，只要 \(P(A) < \delta\)，\(A \in \mathcal{F}\)，则 \(\forall t \in T\)，

\[\int_A |X_t \pm Y_t| dP \leq \int_A |X_t| dP + \int_A |Y_t| dP < \varepsilon. \]

推论 1.1.13

  1. 在 Theorem 1.1.8 中，若 \(\sup_n E|X_n| < +\infty\)，则

\[\sup_n (E|A_n| + E|M_n|) < +\infty. \]

  1. 若 \(\{X_n\}_{n \geq 1}\) 一致可积，则 \(\{M_n\}_{n \geq 1}\)，\(\{A_n\}_{n \geq 1}\) 也一致可积.

证明.

\[E|A_n| = EA_n = \sum_{i=2}^n (EX_i - EX_{i-1}) = EX_n - EX_1 \leq E|X_n| + E|X_1| < +\infty, \]

\[E|M_n| = E|X_n - A_n| \leq E|X_n| + E|A_n| < +\infty. \]

  1. 先证 \(\{A_n\}_{n \geq 1}\) 一致可积：记 \(A = \lim_{n \to \infty} A_n\)，则 \(E|A| = EA\)（因为 \(A \geq 0\)）.

\[0 \leq E|A| = EA = E\left( \lim_{n \to \infty} A_n \right) \leq \liminf_{n \to \infty} EA_n \leq \liminf_{n \to \infty} E|X_n| < \infty. \]

\(\therefore A \in L^1(\Omega, \mathcal{F}, P)\)，且 \(0 \leq A_n \leq A\). 由于 \(E|A| < +\infty\)，于是

\[\lim_{\lambda \to +\infty} \int_{\{|A| > \lambda\}} |A| dP = 0. \]

事实上，\(A I_{\{|A| \geq \lambda\}} \to 0\) a.s. \((\lambda \to +\infty)\)，且 \(\left| A I_{\{|A| \geq \lambda\}} \right| \leq |A|\). 由 DCT：

\[\lim_{\lambda \to +\infty} \int_{\{|A| \geq \lambda\}} |A| dP = 0. \]

\[\int_{\{A_n \geq \lambda\}} A_n dP \leq \int_{\{A_n \geq \lambda\}} A dP \]

\[= \int_{\{A_n \geq \lambda\} \cap \{A \leq k\}} A dP + \int_{\{A_n > \lambda\} \cap \{A > k\}} A dP = I + II. \]

\(I\)：

\[\int_{\{A_n \geq \lambda\} \cap \{A \leq k\}} A dP \leq k P(A_n > \lambda) \leq k \frac{EA_n}{\lambda} \to 0 \quad (\lambda \to +\infty). \]

\(II\)：

\[\int_{\{A_n > \lambda\} \cap \{A > k\}} A dP \xrightarrow{k \to +\infty} 0. \]

综上，\(\{A_n\}_{n \geq 1}\) 是一致可积的。另外，由于 \(M_n = X_n - A_n\)，由 Prop 1.1.12，\(\{M_n\}_{n \geq 1}\) 也一致可积.

Prop 1.1.14 设 \(\{X_n\}_{n \geq 1}\) 一致可积，\(X_n \xrightarrow{\text{a.s.}} X\)，则 \(X_n \xrightarrow{L^1} X\).

证明. 注意

\[E|X| = E|\liminf_{n \to \infty} X_n| \leq \liminf_{n \to \infty} E|X_n| < \infty \]

（因为一致可积）即 \(X \in L^1(\Omega, \mathcal{F}, P)\).

1.2 停时

Def 1.2.1 称随机变量 \(\tau: \Omega \to T \cup \{\infty\}\) 是关于 \(\{\mathcal{F}_t\}_{t \in T}\) 的停时，如果 \(\forall t \in T\)，\(\{\tau \leq t\} \in \mathcal{F}_t\). 若 \(\{\tau < t\} \in \mathcal{F}_t\)，则为宽停时.

Prop 1.2.2

  1. 停时一定为宽停时.

  2. 若 \(\forall t \in T\)，\(\mathcal{F}_t = \mathcal{F}_{t+} \triangleq \bigcap_{s > t} \mathcal{F}_s\)，则宽停时也为停时.

证明.

  1. \(\{\tau < t\} = \bigcup_{n=1}^\infty \{\tau \leq t - \frac{1}{n}\} \in \mathcal{F}_t\).

  2. \(\{\tau \leq t\} = \bigcap_{n=1}^\infty \{\tau < t + \frac{1}{n}\}\)，而 \(\{\tau < t + \frac{1}{n}\} \in \mathcal{F}_{t + \frac{1}{n}}\). 固定 \(m\)，使得 \(\forall n > m\)，有 \(\mathcal{F}_{t + \frac{1}{n}} \subset \mathcal{F}_{t + \frac{1}{m}}\)，则 \(\forall n > m\)，有 \(\{\tau < t + \frac{1}{n}\} \in \mathcal{F}_{t + \frac{1}{m}}\)，故

\[\{\tau \leq t\} \in \bigcap_{m=1}^\infty \mathcal{F}_{t + \frac{1}{m}} \triangleq \mathcal{F}_{t+} = \mathcal{F}_t. \]

Prop 1.2.3 设 \(\tau, \sigma\) 为 \(\{\mathcal{F}_t\}\) 停时，则 \(\tau \wedge \sigma\)，\(\tau \vee \sigma\) 也为停时；若 \(\tau \geq 0\)，\(\sigma \geq 0\)，那么 \(\tau + \sigma\) 也为停时.

证明.

  1. \(\forall t \in T\)，

\[\{\tau \vee \sigma \leq t\} = \{\tau \leq t\} \cap \{\sigma \leq t\} \in \mathcal{F}_t. \]

  1. \(\forall t \in T\)，

\[\{\tau \wedge \sigma \leq t\} = \{\tau \wedge \sigma > t\}^c = \left( \{\tau > t\} \cap \{\sigma > t\} \right)^c = \{\tau \leq t\} \cup \{\sigma \leq t\} \in \mathcal{F}_t. \]

  1. \(\forall t \in T\)，若 \(w \in \{\tau + \sigma \leq t\}\)，即 \(\tau(w) + \sigma(w) \leq t\).

\(\forall n \geq 1\)，\(\exists k \in \{0, 1, \ldots, n\}\)，使得

\[\frac{k t}{n} \leq \tau(w) \leq \frac{(k+1)t}{n}, \quad \sigma(w) \leq t - \frac{k t}{n}. \]

\[\{\tau + \sigma \leq t\} = \bigcap_{n=1}^\infty \bigcup_{k=0}^n \left\{ w : \frac{k t}{n} \leq \tau < \frac{(k+1)t}{n}, \ \sigma \leq t - \frac{k t}{n} \right\} \]

\[\subset \bigcap_{n \in \mathbb{N}} \left\{ w : \tau(w) + \sigma(w) \leq t + \frac{1}{n} \right\} = \{ \tau + \sigma \leq t \}. \]

Def 1.2.4 设 \(\sigma\) 代数流 \(\{\mathcal{F}_t\}_{t \in T}\). 令 \(\mathcal{F}_\infty = \bigvee_{t \in T} \mathcal{F}_t\). 令 \(\tau\) 为停时，定义

\[\mathcal{F}_\tau = \{ B \in \mathcal{F}_\infty \mid \forall t \in T, \ B \cap \{\tau \leq t\} \in \mathcal{F}_t \}. \]

Prop 1.2.5 设 \(\tau\) 为 \(\{\mathcal{F}_t\}_{t \in T}\) 停时，则 \(\tau\) 关于 \(\mathcal{F}_\tau\) 可测.

证明. 我们验证 \(\forall a \in T\)，\(\{\tau \leq a\} \in \mathcal{F}_\tau\). 由于 \(\tau\) 定义，我们要证：
\(\forall t \in T\)，\(\{\tau \leq a\} \cap \{\tau \leq t\} \in \mathcal{F}_t\).

事实上，

\[\{\tau \leq a\} \cap \{\tau \leq t\} = \{\tau \leq a \wedge t\} \in \mathcal{F}_{a \wedge t} \subset \mathcal{F}_t. \]

所以 \(\tau\) 关于 \(\mathcal{F}_t\) 可测.

Prop 1.2.6 若 \(\tau\) 是 \(\{\mathcal{F}_t\}_{t \in T}\) 停时，\(\sigma\) 也是 \(\{\mathcal{F}_t\}_{t \in T}\) 停时，且几乎处处成立 \(\tau \leq \sigma\)，则 \(\mathcal{F}_\tau \subset \mathcal{F}_\sigma\).

证明. \(\forall A \in \mathcal{F}_\tau\)，由定义，\(\forall t \in T\)，\(A \cap \{\tau \leq t\} \in \mathcal{F}_t\).

\[A \cap \{\sigma \leq t\} = \underbrace{A \cap \{\tau \leq t\}}_{\in \mathcal{F}_t} \cap \underbrace{\{\sigma \leq t\}}_{\in \mathcal{F}_t} \in \mathcal{F}_t. \]

故 \(A \in \mathcal{F}_\sigma\).

Prop 1.2.7 设 \(\{X_t\}_{t \in T}\) 关于 \(\{\mathcal{F}_t\}_{t \in T}\) 适应，\(\tau: \Omega \to \mathbb{Z}_+\) 是停时，且 \(\tau\) 只取有限值，则 \(X_\tau\) 关于 \(\mathcal{F}_\tau\) 可测.

证明. \(\forall B \in \mathcal{B}(\mathbb{R})\)（假设 \(X_t: (\Omega, \mathcal{F}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))\)），

\[\{X_\tau \in B\} \cap \{\tau \le n\} = \sum_{k=1}^n \{X_\tau \in B\} \cap \{\tau = k\}=\sum_{k=1}^n \{X_k \in B\} \in \mathcal{F}_n. \]

故 \(\{X_\tau \in B\} \in \mathcal{F}_\tau\).

Thm 1.2.8 (停时定理) 设\(\{ X_n, \mathcal{F}_n \}_{n\in Z^+}\), \(\sigma, \tau\)是有界停时，而且\(\sigma \le \tau\), 则\(E(X_tau |\mathcal{F}_\sigma)=X_\sigma\), a.s.

Pf. 根据条件期望的定义，我们只需要证明：\(\forall A \in \mathcal{F}_\sigma, E(X_\tau I_A)=E(X_\sigma I_A)\). 根据有界性条件，我们有 \(\sigma \leqslant \tau \leqslant m\)，则

\[\begin{aligned} E(X_\tau I_A) &= \sum_{k=1}^m E\left[X_\tau I_{A \cap \{\tau = k\}}\right] \\ &= \sum_{k=1}^m E\left[X_k I_{A \cap \{\tau = k\}}\right] \\ &= \sum_{k=1}^m E\left[E(X_m | \mathcal{F}_k) I_{A \cap \{\tau = k\}}\right] \\ &= \sum_{k=1}^m E\left(E(X_m I_{A \cap \{\tau = k\}} | \mathcal{F}_k)\right) \quad \text{由于 } A \in \mathcal{F}_\tau, \text{ 所以 } A \cap \{\tau = k\} \in \mathcal{F}_k \\ &= \sum_{k=1}^m E\left(X_m I_{A \cap \{\tau = k\}}\right) \\ &= E(X_m I_A). \end{aligned}\]

类似有 \(E(X_\sigma I_A) = E(X_m I_A)\)，所以 \(E(X_\tau | \mathcal{F}_\sigma) = X_\sigma\) a.s.

posted @ 2026-08-08 19:03  夜秋子  阅读(11)  评论(0)    收藏  举报