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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: \[ f_{M,W}(m,w)=\frac{2(2m-w)}{t \sqrt{2\pi t}}e^{- \frac{(2m-\omega)^2}{2t}} \]