随机过程 | 鞅论 Doob下鞅分解定理

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 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，\(E[\varphi(X_t)] < +\infty\).

2. \(\varphi(X_t) \in \mathcal{F}_t\), \(E|\varphi(t)| < +\infty\)（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.

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

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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 \(P(A) < \delta\)，\(\int_A |X_t| dP < \varepsilon\)