布朗运动与随机微积分

5.4.1 布朗运动

1. 定义：过程 $\{W(t),t\ge 0\}$ 是标准布朗运动若

• $W(0)=0$；
• 增量独立；
• 增量 $W(t_{i+1})-W(t_i)\sim N(0,t_{i+1}-t_i)$；
• 样本路径连续（无跳跃）。

2. 性质：

• $W(t)\sim N(0,t)$，因此 $\mathbb{E}[W(t)]=0$、$\mathrm{Var}(W(t))=t$。
• 鞅性质：$\mathbb{E}[W(t+s)\mid W(t)]=W(t)$。
• Markov 性。
• $\operatorname{cov}(W(s),W(t))=\min(s,t)$。

3. 相关鞅：

• $Y(t)=W(t)^2-t$ 是鞅。
• 指数鞅：

$$Z(t)=\exp\{\lambda W(t)-\tfrac12\lambda^2 t\}.$$

4. $\operatorname{Cov}(B_t,B_t^2)=0$：因为 $B_t\sim N(0,t)$，有 $\mathbb{E}[B_t^3]=0$，于是

$$\operatorname{Cov}(B_t,B_t^2)=\mathbb{E}[B_t^3]-\mathbb{E}[B_t]\mathbb{E}[B_t^2]=0.$$

5. $P(B_1>0,B_2<0)$：由对称性（或反射原理）可得该概率为 $1/8$。

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5.4.2 停时 / 首次穿越时间

• 命中 $\pm 1$：令 $T=\min\{t\mid B_t=1\text{ 或 }-1\}$。对鞅 $M(t)=B_t^2-t$ 用可选停时定理：

$$\mathbb{E}[B_T^2-T]=\mathbb{E}[B_0^2]=0.$$

且 $B_T^2=1$，因此 $\mathbb{E}[T]=1$。

• 一般命中 $+\alpha$ 或 $-\beta$（$\alpha,\beta>0$）时：$\mathbb{E}[T]=\alpha\beta$。

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5.4.3 Ito 引理

若

$$dX_t=\beta(t,X_t)\,dt+\gamma(t,X_t)\,dW_t,$$

且 $f(t,x)$ 足够光滑，则

$$df(t,X_t)=\Bigl(\partial_t f+\beta\,\partial_x f+\tfrac12\gamma^2\,\partial_{xx} f\Bigr)dt+\gamma\,\partial_x f\,dW_t.$$

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Original Explanation

5.4.1 Brownian Motion

1. Definition: A process $\{W(t),t\ge 0\}$ is a standard Brownian motion if:

   • $W(0)=0$,
   • It has independent increments,
   • Each increment $W(t_{i+1})-W(t_i)$ is normally distributed with mean 0, variance $t_{i+1}-t_i$,
   • It has continuous paths (no jumps).

2. Properties:

   • $W(t)\sim N(0,t)$, hence $\mathbb{E}[W(t)]=0$, $\mathrm{Var}(W(t))=t$.
   • Martingale: $\mathbb{E}[W(t+s)\mid W(t)] = W(t)$.
   • Markov property.
   • $\operatorname{cov}(W(s),W(t))=\min(s,t)$.

3. Martingales related to BM:

   • $Y(t)=W(t)^2 - t$ is a martingale.
   • The exponential martingale: $$Z(t) = \exp\bigl\{\lambda\,W(t) - \tfrac12\,\lambda^2\,t\bigr\}.$$

4. $\operatorname{Cov}(B_t,B_t^2)=0$. Since $B_t\sim N(0,t)$, $\mathbb{E}[B_t^3]=0$. Then $$\operatorname{Cov}(B_t,B_t^2) = \mathbb{E}[B_t^3] - \mathbb{E}[B_t]\mathbb{E}[B_t^2] = 0 - 0\cdot t = 0.$$

5. Probability $\{B_1>0,B_2<0\}$ is $1/8$. By symmetry (or reflection principle arguments), the chance that $B_1>0$ and later $B_2<0$ ends up being $1/8$.

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5.4.2 Stopping Time / First Passage Time

• For Brownian motion hitting $\pm 1$ for the first time:

  • Let $T = \min\{\,t\mid B_t=1 \text{ or } B_t=-1\}$. Using the martingale $M(t)=B_t^2 - t$, we get $\mathbb{E}[B_T^2 - T]=\mathbb{E}[B_0^2]=0$. But $B_T^2=1$. Thus $\mathbb{E}[T]=1$.

• In general, to hit +$\alpha$ or -$\beta$ ($\alpha,\beta>0$), $\mathbb{E}[T] = \alpha\,\beta$.

• With drift $m$, $dX_t = m\,dt + dW_t$, one solves via ODE or exponential martingale. For example, from 0 to +3 or -5: $$p_{\,+3} = \frac{e^{10\,m}-1}{\,e^{10\,m}-e^{-6\,m}\,}.$$ If $m=0$, it becomes $\tfrac58$.

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5.4.3 Ito’s Lemma

Ito’s lemma says for $$dX_t = \beta(t,X_t)\,dt + \gamma(t,X_t)\,dW_t,$$ and $f(t,x)$ sufficiently smooth, $$df = \bigl[\partial_t f + \beta\,\partial_x f + \tfrac12\,\gamma^2\,\partial_{xx} f\bigr]\,dt + \gamma\,\partial_x f\;dW_t.$$

• Example: $Z_t = \sqrt{t}\,B_t$. By Ito’s lemma, it has a drift term $\tfrac12\,t^{-1/2}\,B_t$, so it is not a martingale.

• Example: $B_t^3$. The drift term is $3\,B_t\,dt$, so $B_t^3$ is not a martingale.