Brownian Motion

Brownian motion (or the Wiener process) is the most fundamental continuous-time stochastic process, characterized by independent Gaussian increments and continuous paths. It is the cornerstone of stochastic calculus, mathematical finance, and diffusion theory.

Definition

Definition4.1Standard Brownian motion

A stochastic process $(B_t)_{t \geq 0}$ is a standard Brownian motion if:

$B_0 = 0$ almost surely.
Independent increments: For any $0 \leq t_1 < t_2 < \cdots < t_n$ , the increments $B_{t_2}-B_{t_1}, B_{t_3}-B_{t_2}, \ldots, B_{t_n}-B_{t_{n-1}}$ are independent.
Gaussian increments: For $s < t$ , $B_t - B_s \sim \mathcal{N}(0, t-s)$ .
Continuous paths: $t \mapsto B_t(\omega)$ is continuous for a.e. $\omega$ .

Properties (2-3) imply $(B_t)$ is a Gaussian process with mean 0 and covariance $\mathbb{Cov}(B_s, B_t) = \min(s,t)$ .

ExampleScaling and self-similarity

For any $c > 0$ , the process $(B_{ct})_{t \geq 0}$ has the same distribution as $(\sqrt{c} B_t)_{t \geq 0}$ . This scale invariance is a defining feature of Brownian motion.

Construction and existence

Theorem4.1Wiener's existence theorem

There exists a probability space carrying a standard Brownian motion.

Proof outline: Use the Lévy-Ciesielski construction by defining $B_t$ on dyadic rationals, then extending by uniform continuity.

Properties

Theorem4.2Martingale property

$(B_t)$ is a martingale: $\mathbb{E}[B_t \mid \mathcal{F}_s] = B_s$ for $s < t$ .
$(B_t^2 - t)$ is a martingale.
$(\exp(\theta B_t - \frac{1}{2}\theta^2 t))$ is a martingale for any $\theta \in \mathbb{R}$ .

ExampleStock prices

In the Black-Scholes model, a stock price follows $S_t = S_0 \exp(\mu t + \sigma B_t)$ , where $B_t$ is Brownian motion with drift $\mu$ and volatility $\sigma$ .

Summary

Brownian motion has independent Gaussian increments, continuous paths, and the martingale property. It is the continuous-time limit of random walks and the foundation for Itô calculus.