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Chapter 1 Brownian Motion

Brownian motion at time t is limit of infinite fast random walk W^{n}_t, it can be equivalently characterized by

  1. Any increment W_{t_1}-W_{t_2} is normal distributed with mean 0 and variance t_1-t_2. Disjoint increment are independency.
  2. For any time t_1,t_2,\dots,t_{m}, \bm{W}=(W_{t_1},W_{t_2},\dots,W_{t_{m}}) is normal distributed with zero mean and covariance \bm{\Sigma}=\begin{bmatrix} t_1 & t_1 & \dots & t_1 \\ t_1 & t_2 & \dots & t_2 \\ \vdots & \vdots & & \vdots \\ t_1 & t_2 & \dots & t_{m} \\ \end{bmatrix}
  3. \bm{W} has MGF \varphi(\bm{t})=\exp \left\{ \sum_{i=1}^{m}\frac{1}{2}\left( \sum_{j=i}^{m} t_{j} \right)^2 (t_i-t_{i-1}) \right\} where t_0=0.

Brownian motion is a Markov martingale with variation:

  1. dW_{t}=\infty
  2. dW_t dW_{t}=dt
  3. dW_{t}dt=dt^2=0

1.1 Markov Property

Lemma 1.1 (Independence Lemma) Suppose X\in \mathcal{A}, Y \perp \mathcal{A}, then \mathop{{}\mathbb{E}}_{\mathcal{A}}f(X,Y)=\mathop{{}\mathbb{E}}_{}f(x,Y)|_{x=X}

Proof. When f=g\times h for some g,h, then \mathop{{}\mathbb{E}}_{\mathcal{A}}f(X,Y)=\int K(X,dy)f(X,y)=\int \mu(dy)f(X,y)=\mathop{{}\mathbb{E}}_{}f(x,Y)|_{x=X} since product \sigma algebra is generated by measurable rectangles, monotone class theorem completes the proof.

Preceding lemma implies \begin{aligned} \mathop{{}\mathbb{E}}_{s}f(W_{t})&=\mathop{{}\mathbb{E}}_{s}f(W_{t}-W_{s}+W_{s}) \\ &= \mathop{{}\mathbb{E}}_{}f(W_{t}-W_{s}+x)|_{x=W_{s}} \\ &= \frac{1}{\sqrt{2\pi(t-s)}}\int_{\mathbb{R}} f(w+x)\exp \left\{ - \frac{w^2}{2(t-s)} \right\}dw|_{x=W_{s}} \\ &\xlongequal{\tau=t-s,y=w+x} \frac{1}{\sqrt{2\pi \tau}}\int_{\mathbb{R}} f(y)\exp \left\{ - \frac{w^2}{2\tau} \right\}dw|_{x=W_{s}} \\ & \xlongequal{} \int_{\mathbb{R}}f(y)p(\tau,W_{s},y)dy \end{aligned} where p(\tau,W_{s},y) is pdf of \mathcal{N}(W_{s},\tau).

1.2 Exponential Martingale

Proposition 1.1 () Suppose W_t is a Brownian Motion with filtration \mathbb{F}, then process Z_t=\exp \left\{ \sigma W_t - \frac{1}{2} \sigma^2t\right\} is a martingale.

Define the first passage time to m as \tau_m=\min \left\{ t\ge 0,W_t=m \right\} recall the stopped martingale, we have 1=\mathop{{}\mathbb{E}}_{}Z_0=\mathop{{}\mathbb{E}}_{}Z_{t \land \tau_m}=\mathop{{}\mathbb{E}}_{}\exp \left\{ \sigma W_{t \land \tau_m} - \frac{1}{2} \sigma^2(t \land m) \right\} Taking limit inside expectations: \lim_{t \to \infty} \exp \left\{ \sigma W_{t \land \tau_m} - \frac{1}{2} \sigma^2(t \land m) \right\}=\bm{1}_{\tau_{m}<\infty}\exp \left\{\sigma m -\frac{1}{2} \sigma^2 \tau_{m} \right\} that implies \mathop{{}\mathbb{E}}_{}\bm{1}_{\tau_{m}<\infty} \exp \left\{ -\frac{1}{2} \sigma^2 \tau_{m} \right\}= \exp \left\{ -\sigma m \right\} take \sigma \searrow 0, we have \tau_{m} is finite a.s..

And the characteristic function of \tau_{m} is given by taking t=\frac{1}{2}\sigma^2: \mathop{{}\mathbb{E}}_{}\exp \left\{ -t \tau_{m} \right\}=\exp \left\{ - \left| m \right|\sqrt{2 t} \right\}

1.3 Reflection

By the symmetry of Brownian motion, we have \mathop{{}\mathbb{P}}\left\{ \tau_{m}\le t,W_t\le w \right\}=\mathop{{}\mathbb{P}}\left\{ W_t\ge 2m-w \right\} when 0<m\ge w. On the other hand: \mathop{{}\mathbb{P}}\left\{ \tau_{m}\le t,W_t\ge w \right\}=\mathop{{}\mathbb{P}}\left\{ W_t\ge w \right\} take m=w and adding these two: \mathop{{}\mathbb{P}}\left\{ \tau_{m}\le t \right\}=2\mathop{{}\mathbb{P}}\left\{ W_{t} \ge m \right\}

1.3.1 Joint Distribution of Brownian Motion and its maximum

Define maximum process: M_{t}=\max_{0\le s\le t}W_{s} clearly, M_{t}\ge m iff \tau_{m}\le t, thus \mathop{{}\mathbb{P}}\left\{ M_{t}\ge m,W_{t}\le w \right\}=\mathop{{}\mathbb{P}}\left\{ W_{t}\ge 2m-w \right\} from which we have: f_{M,W}(m,w)=\frac{2(2m-w)}{t \sqrt{2\pi t}}e^{- \frac{(2m-\omega)^2}{2t}}