Brownian Motion

Brownian Motion

A stochastic process $(B_t{)}_{t\in [0,1]}$ is called a standard Brownian motion on $[0,1]$ if it has the following properties:

1. $B_0=0$.
2. For any set of finite times, $0\le t_1<t_2<\cdots <t_N\le 1$, the increments of the process $(B_t{)}_{t\in [0,1]}$, $B_{t_2}-B_{t_1},B_{t_3}-B_{t_2},\dots ,B_{t_N}-B_{t_{N-1}},$ are all independent.
3. $B_t-B_s\sim \mathcal{N}(0,t-s)$ for any $s,t\in [0,1]$ with $s\le t$.
4. The paths of $B_t$ are almost surely continuous. That is, $\mathbb{P}(t\mapsto B_t\text{ is continuous})=1.$

Theorem

Let $(Z_n{)}_{n\in \mathbb{N}}$ be a sequence of independent Gaussian variables with $Z_n\sim \mathcal{N}(0,1)$ for each $n\in \mathbb{N}$. The series $X_t:={\sum }_{n=0}^{\infty }⟨H_n,1_{[0,t]}⟩Z_n$ converges uniformly on $[0,1]$ almost surely. Moreover, the process $(X_t{)}_{t\in [0,1]}$ is a standard Brownian motion on $[0,1]$.

Standard Brownian Motion

Let $(B_t^{(j)}{)}_{t\in [0,1]}$ be independent standard Brownian motions on $[0,1]$ for $j\in \mathbb{N}$. Define for $n\in \mathbb{N}$ and $s\in [n,n+1)$, $W_s:={\sum }_{j=0}^{n-1}{B}_1^{(j)}+B_{s-n}^{(n)}.$ $(W_s{)}_{s\ge 0}$ is called a standard Brownian motion on $[0,\infty )$.

Continuous Time Filtration

A family of sigma-algebras $({\mathcal{F}}_t{)}_{t\ge 0}$ is called a filtration for the probability space $(\Omega ,\Sigma ,\mathbb{P})$ if for any $t,s\ge 0$ with $t\ge s$ we have ${\mathcal{F}}_s\subset {\mathcal{F}}_t\subset \Sigma .$ A process $(X_t{)}_{t\ge 0}$ is said to be adapted to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ if $X_t$ is ${\mathcal{F}}_t$-measurable for each $t\ge 0$.

Standard (augmented) Brownian Filtration

Let $(W_s{)}_{s\ge 0}$ be a standard Brownian motion on $[0,\infty )$. Define, for each $t\ge 0$, ${\mathcal{F}}_t^W:=\sigma (W_s:s\in [0,t])$ ${\mathcal{F}}_t^{W,+}:={\bigcap }_{s>t}{\mathcal{F}}_s^W{\mathcal{N}}_t:=\{A\in \Sigma :A\subset B\text{ for some }B\in {\mathcal{F}}_t\text{ with }\mathbb{P}(B)=0\}.$ The standard (augmented) Brownian filtration, $({\mathcal{F}}_t{)}_{t\ge 0}$, is defined by ${\mathcal{F}}_t:=\sigma \left({\mathcal{F}}_t^{W,+}\cup {\mathcal{N}}_t\right)\forall t\ge 0.$

Continuous Martingale

Let $(X_t{)}_{t\ge 0}$ be a process with $X_t\in L^2(\Omega ,{\mathcal{F}}_t,\mathbb{P})$ for each $t\ge 0$. $(X_t{)}_{t\ge 0}$ is called a martingale with respect to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ if for any $s,t\ge 0$ with $s\le t$ $\mathbb{E}(X_t\mid {\mathcal{F}}_s)=X_s.$ The martingale is called continuous if it almost surely has continuous paths. That is, $\mathbb{P}(t\mapsto X_t\text{ continuous})=1.$

Stopping Time

A random variable $\tau :\Omega \to [0,\infty ]$ is called a stopping time with respect to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ if $\{\tau \le t\}\in {\mathcal{F}}_t$ for any $t\ge 0$.

Uniformly Integrable

A collection $\mathcal{C}$ of random variables in $L^1(\Omega ,\Sigma ,\mathbb{P})$ is called uniformly integrable if ${\lim }_{K\to \infty }{\sup }_{Z\in \mathcal{C}}\mathbb{E}\left(|Z|1_{|Z|\ge K}\right)=0.$

Proposition

Let $(Z_n{)}_{n\in \mathbb{N}}$ be uniformly integrable and $Z_n\to Z$ almost surely as $n\to \infty$. Then $Z\in L^1(\Omega ,\Sigma ,\mathbb{P})$ and $\Vert Z_n-Z{\Vert }_{L^1}\to 0$ as $n\to \infty$.

Theorem

Let $({\mathcal{F}}_t{)}_{t\ge 0}$ be a filtration and $(M_t{)}_{t\ge 0}$ be a continuous martingale with respect to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$. Let $\tau$ be a stopping time with respect to $({\mathcal{F}}_t{)}_{t\ge 0}$. Then the stopped process $(M_{t\wedge \tau }{)}_{t\ge 0}$ is a continuous martingale with respect to $({\mathcal{F}}_t{)}_{t\ge 0}$.

Doob's Maximal and $L^p$ Inequalities

Let $T>0$ and $(M_t{)}_{t\ge 0}$ be a continuous submartingale such that $M_t\ge 0$ almost surely for all $t\in [0,T]$. Then for any $\alpha >0$, $\alpha \cdot \mathbb{P}\left({\sup }_{0\le t\le T}{M}_t>\alpha \right)\le \mathbb{E}(|M_T|)$ and for any $p\in (1,\infty )$, ${\Vert {\sup }_{0\le t\le T}{M}_t\Vert }_p\le \frac{p}{p-1}\Vert M_T{\Vert }_p.$

Martingale Convergence Theorem

Let $(M_t{)}_{t\ge 0}$ be a continuous martingale with respect to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$. If ${\sup }_{t\ge 0}\mathbb{E}\left(|M_t{|}^2\right)<\infty$ then there exists $M_{\infty }\in L^2(\Omega ,\Sigma ,\mathbb{P})$ such that $M_t$ converges to $M_{\infty }$ almost surely and in the $L^2$-norm.

Proposition

Let $A,B>0$. Define $\tau :\Omega \to [0,\infty ]$ by $\tau (\omega ):=\inf \{t\ge 0:B_t=-B\text{ or }{B}_t=A\}$ where we take $\tau (\omega )=\infty$ if these values are never reached. Then: (i) $\mathbb{P}(\tau <\infty )=1$. (ii) $\mathbb{P}(B_{\tau }=A)=\frac{B}{A+B}$. (iii) $\mathbb{E}(\tau )=A\cdot B<\infty$.