本章导读 本章引入停时并证明布朗运动的强马尔可夫性。§5.1 定义与例（Def 5.1 停时：\(\{\tau\le t\}\in\mathcal F_t\)；Rmk 5.1 直觉——停时只依赖到关心时刻为止的信息、从不涉及未来；Ex 5.1 例子：首达时、常数、两停时的 \(\min/\max\)；Def 5.2 停时 \(\sigma\)-代数 \(\mathcal F_\tau\)）。§5.2 强马尔可夫性（Thm 5.1：在停时 \(\tau\) 处平移去基点后 \(\{Y_t\}=\{B_{t+\tau}-B_\tau\}\) 仍是布朗运动且独立于 \(\mathcal F_\tau\)；证明先对有限取值 \(\tau\) 用普通马氏性、再用二进网格逼近；Rmk 5.2、5.3：与普通马氏性的区别仅在 \(\tau\) 是随机变量）。无图。

定义 5.1（停时）/ Definition 5.1 (Stopping time) 随机变量 \(\tau:\Omega\to(0,\infty)\) 称为关于滤波 \(\{\mathcal F_t\}\) 的停时，若对 \(\forall t\) 有A random variable \(\tau:\Omega\to(0,\infty)\) is called a stopping time w.r.t. a filtration \(\{\mathcal F_t\}\) if for all \(t\)

$$\{\tau\le t\}\in\mathcal F_t.$$

注 5.1 / Remark 5.1 5.1 的直觉是：停时永远只依赖到我们所关心时刻为止的信息，而绝不涉及该时刻之后（未来）的信息。The intuitive explanation of 5.1 is that a stopping time always depends only on information up to the time we are interested in, and never involves information about the future of that point.

例 5.1（停时的例子）/ Example 5.1 (Stopping time) 以下都是停时：(1) 首达时 \(\tau_a=\min\{t:B_t=a\}\)；(2) 常数 \(\tau=C\)（平凡）；(3) 若 \(\tau_1,\tau_2\) 都是停时，则 \(\tau_1\wedge\tau_2\) 与 \(\tau_1\vee\tau_2\) 也是停时（脚注 5.1：\(\tau_1\wedge\tau_2\) 表较小者、\(\tau_1\vee\tau_2\) 表较大者）。The following are stopping times: (1) the first-hitting time \(\tau_a=\min\{t:B_t=a\}\); (2) a constant \(\tau=C\) (trivial); (3) if \(\tau_1,\tau_2\) are both stopping times, so are \(\tau_1\wedge\tau_2\) and \(\tau_1\vee\tau_2\) (footnote 5.1: \(\tau_1\wedge\tau_2\) means the smaller of \(\tau_1,\tau_2\), and \(\tau_1\vee\tau_2\) the greater).

定义 5.2（停时 σ-代数 \(\mathcal F_\tau\)）/ Definition 5.2 (Stopping-time σ-algebra) 若 \(\tau\) 是停时，则 \(\mathcal F_\tau\) 是所有满足如下条件的事件 \(A\) 构成的 \(\sigma\)-代数：对 \(\forall t\) 有 \(A\cap\{\tau\le t\}\in\mathcal F_t\)。If \(\tau\) is a stopping time, then \(\mathcal F_\tau\) is the \(\sigma\)-algebra of all events \(A\) such that for all \(t\), \(A\cap\{\tau\le t\}\in\mathcal F_t\).

定理 5.1（强马尔可夫性）/ Theorem 5.1 (Strong Markov Property) 设 \(\{B_t\}\) 是布朗运动，\(\tau\) 是满足 \(\mathbb P\{\tau<\infty\}=1\) 的停时。令 \(Y_t=B_{t+\tau}-B_\tau\)（\(t\ge0\)）。则 \(\{Y_t\}\) 是一个独立于 \(\mathcal F_\tau\) 的布朗运动。Let \(\{B_t\}\) be a Brownian motion and \(\tau\) a stopping time with \(\mathbb P\{\tau<\infty\}=1\). Let \(Y_t=B_{t+\tau}-B_\tau\) for \(t\ge0\). Then \(\{Y_t\}\) is a Brownian motion independent of \(\mathcal F_\tau\).

定理 5.1 证明 / Proof of Theorem 5.1 第一步（有限取值）：先假设 \(\tau\) 只取有限个值，即把样本空间划分为 \(\{\tau=s_1\}\cup\{\tau=s_2\}\cup\dots\cup\{\tau=s_k\}\)。在每个 \(\{\tau=s_i\}\) 上 \(\tau\) 退化为常数 \(s_i\)，于是可用普通马尔可夫性（命题 4.1）逐块论证。第二步（一般情形）：对一般的 \(\tau\)，用二进网格逼近并取极限。令Step 1 (finitely-valued): first assume \(\tau\) takes only a finite number of values, i.e. partition the space into \(\{\tau=s_1\}\cup\{\tau=s_2\}\cup\dots\cup\{\tau=s_k\}\). On each \(\{\tau=s_i\}\), \(\tau\) degenerates to the constant \(s_i\), so we can use the ordinary Markov property (Proposition 4.1) to argue on each. Step 2 (general case): for a more general \(\tau\), approximate it by dyadic grids and take limits. Let

$$\tau_n\in\left\{\frac1{2^n},\frac2{2^n},\dots,\frac{n\cdot2^n}{2^n}\right\}$$

并设 \(\tau_n=\dfrac{j}{2^n}\) 当 \(\left\{\dfrac{j-1}{2^n}\le\tau\le\dfrac{j}{2^n}\right\}\)，且设 \(\tau_n=n\) 当 \(\left\{\tau\ge\dfrac{n\cdot2^n-1}{2^n}\right\}\)。然后令 \(n\to\infty\) 取极限，\(\tau_n\) 即可逼近一般情形。\(\blacksquare\)and set \(\tau_n=\dfrac{j}{2^n}\) if \(\left\{\dfrac{j-1}{2^n}\le\tau\le\dfrac{j}{2^n}\right\}\), and set \(\tau_n=n\) if \(\left\{\tau\ge\dfrac{n\cdot2^n-1}{2^n}\right\}\). Then take the limit \(n\to\infty\) to let \(\tau_n\) approximate the general case. \(\blacksquare\)

注 5.2 与注 5.3 / Remarks 5.2 and 5.3 注 5.2：强马尔可夫性的直觉是——过程停止之后发生的一切，独立于停止之前发生的一切。注 5.3：强马尔可夫性看起来与普通马尔可夫性几乎一样，区别仅在于这里的 \(\tau\) 也是一个随机变量，而普通情形下它只是一个数 \(t_0\)。Remark 5.2: the intuition of the strong Markov property is that everything happening after the process stops is independent of everything before stopping. Remark 5.3: the strong Markov property looks almost exactly the same as the ordinary Markov property, except that here \(\tau\) is also a random variable, whereas in the ordinary case it is just a number \(t_0\).