Brownian Motion and Its Properties

You must have chaos within you to give birth to a dancing star.
Friedrich Nietzsche

The first time when I heard the term Brownian motion is from a physics class and it was introduced as the random collisions of particles. In the field of Stochastic Differential Equations, it represents a stochastic process.

Brownian Motion:

Definition 1. A real-valued stochastic process $W(\cdot)$ is called a Brownian motion or Wiener process if

$W(0)=0$ almost surely.
$W(t)-W(s)$ is $N(0,t-s)$ for all $t \geq s > 0$ .
For all time $0 <t_1<t_2<\dots<t_n$ , the random variables $W(t_1),W(t_2)-W(t_1),\dots,W(t_n)-W(t_{n-1})$ are independent (independent increments).

Note that the construction of Brownian motion is really careful, and it is important to demonstrate the existence of a Brownian motion. One way is to use an orthonormal
basis of $L^2(0,1)$ and prove the convergence of a resulting series. One can consult the reference for further study, and here I will present the final result.

Theorem 1 (Existence of one-dimensional Brownian motion). Let $(\Omega,\mathcal{U},P)$ be a probability space on which countably many $N(0,1)$ , independent random variables $\{A_n\}_{n=1}^\infty$ are defined. Then there exists a 1-dimensional Crownian motion $W(\cdot)$ defined for $\omega \in \Omega$ , $t \geq 0$ .

The existence of Brownian motion in $\mathbb{R}^n$ can then be extended from the one-dimensional case.

Now we will take a look at the properties of the sample path of Brownian motions. First, recall the definition of Hölder continuity.

Definition 2. Let $0 < \gamma \leq 1$ . A function $f:[0,T] \to \mathbb{R}$ is called Hölder continuous with exponent $\gamma >0$ if there exists a constant $K$ such that

$|f(t)-f(s)| \leq K|t-s|^\gamma \quad \text{for all }t \in [0,T]$

We say $f$ is uniformly Hölder continuous with exponent $\gamma >0$ if there exists a constant $K$ such that

$|f(t)-f(s)| \leq K|t-s|^\gamma \quad \text{for all }s,t \in [0,T]$

The continuity of sample paths can be established by a theorem of Kolmogorov

Theorem 2. Let $X(\cdot)$ be a stochastic process with continuous sample paths a.s., such that

$E(|X(t)-X(s)|^\beta) \leq C|t-s|^{1+\alpha}$

for constants $\beta,\alpha >0$ , $C \geq 0$ and for all $0 \leq t,s$ .

Then for each $0<\gamma<\alpha/\beta$ , $T>0$ , and almost every $\omega$ , there exists a constant $K=K(\omega,\gamma,T)$ such that

$|X(t,\omega)-X(s,\omega)| \leq K|t-s|^\gamma \quad\text{for all }0\leq s,t\leq T$

Hence the sample path $t \mapsto X(t,\omega)$ is uniformly Hölder continuous with exponent $\gamma$ on $[0,T]$ .

And we have the following result

Theorem 3. For almost every $\omega$ and any $T>0$ , the sample path $t \mapsto X(t,\omega)$ is uniformly Hölder continuous on $[0,T]$ for each exponent $0<\gamma<1/2$ . For each $1/2<\gamma\leq 1$ and almost every $\omega$ , the sample path $t \mapsto X(t,\omega)$ is nowhere Hölder continuous with exponent $\gamma$ . In particular, for almost every $\omega$ , the sample path $t \mapsto X(t,\omega)$ is nowhere differentiable and is of infinite variation on each subinterval.

These properties are important because they also guarantee the continuity of indefinite Ito’s integral and the continuity of some other applications like the solutions of stochastic differential equations.

Another property of Brownian motion is the Markov property, as represented below.

Definition 3. If $X(\cdot)$ is a stochastic process, the $\sigma$ -algebra

$\mathcal{U}(s) = \mathcal{U}(s)(X(r)|0\leq r \leq s )$

is called the history of the process up to and including time $s$ .

Definition 4. If $\mathcal{V}$ is a $\sigma$ -algebra, $\mathcal{V} \subseteq \mathcal{U}$ , then

$P(A|\mathcal{V}) = E(\chi_A|\mathcal{V}) \quad \text{for }A \in \mathcal{U}$

is a random variable and defined as the conditional probability of $A$ given $\mathcal{V}$ .

Definition 5. An $\mathbb{R}^n$ -valued stochastic process $X(\cdot)$ is called a Markov process if

$P(X(t) \in B|\mathcal{U}(s)) = P(X(t) \in B|X(s))\quad a.s.$

for all $0 \leq s \leq t$ and all Borel subset $B$ of $\mathbb{R}^n$ .

And now we have the following theorem:

Theorem 4. Let $W(\cdot)$ be an $n$ -dimensional Brownian motion. Then $W(\cdot)$ is a Markov process, and

$P(W(t) \in B|W(s)) = \frac{1}{(2\pi (t-s))^{n/2}} \int_B \exp \left\{-\frac{|x-W(s)|^2}{2(t-s)} \right\}\quad a.s.$

for all $0 \leq s <t$ , and Borel sets $B$ .

Note that one can also prove the strong Markov property for Brownian motion. It helps to define the harmonic measure of an Ito diffusion and further find the solution to the generalized Dirichlet problem.

Reference

[1] Øksendal, Bernt K. (2013). Stochastic differential equations: An introduction with applications. Heidelberg: Springer.

[2] Evans, Lawrence C. An Introduction to Stochastic Differential Equations. Department of Mathematics, UC Berkeley.

Brownian Motion and Its Properties

Brownian Motion:

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Brownian Motion:

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