Brownian Motion & Wiener Process
Brownian Motion & Wiener Process
Brownian motion (Wiener process) $\{W_t\}_{t\ge 0}$ is the fundamental continuous-time, continuous-path stochastic process. It arises as the scaling limit of random walks and drives virtually all of continuous-time finance and physics. Its almost-sure path properties — continuity but nowhere differentiability — make it analytically challenging and demand the Ito stochastic calculus developed in Lesson 6.
Brownian motion is fully characterized by its Gaussian finite-dimensional distributions: $(W_{t_1},\ldots,W_{t_n})$ is jointly normal with $E[W_t]=0$ and $\text{Cov}(W_s,W_t)=\min(s,t)$. This covariance structure encodes the independent increments property.
Standard Brownian Motion
A process $\{W_t\}_{t\ge 0}$ is a standard Brownian motion if: (i) $W_0=0$ a.s.; (ii) independent increments: $W_{t}-W_s\perp\mathcal{F}_s$ for $t>s$; (iii) stationary Gaussian increments: $W_t-W_s\sim\mathcal{N}(0,t-s)$; (iv) continuous paths a.s. Equivalently, $W$ is a Gaussian process with $E[W_t]=0$ and $\text{Cov}(W_s,W_t)=s\wedge t$.
Path Properties (Lévy)
Brownian paths are a.s. Hölder continuous of any exponent $\alpha<1/2$, but a.s. nowhere $1/2$-Hölder and nowhere differentiable. The quadratic variation satisfies $[W]_t=t$ a.s. (i.e., $\sum_{k}(W_{t_{k+1}}-W_{t_k})^2\to t$ in $L^2$ as the mesh $\to 0$), which replaces the classical chain rule and motivates Ito's formula.
Example 1 — Reflection Principle
For $a>0$, $P(\max_{0\le s\le t}W_s\ge a)=2P(W_t\ge a)=2\Phi(-a/\sqrt{t})$, where $\Phi$ is the standard normal CDF. This follows by reflecting the path at its first hitting time of $a$.
Example 2 — Geometric Brownian Motion
$S_t=S_0\exp\!(\mu t+\sigma W_t)$ is the canonical model for stock prices. Then $E[S_t]=S_0 e^{(\mu+\sigma^2/2)t}$ and $\text{Var}(S_t)=S_0^2 e^{(2\mu+\sigma^2)t}(e^{\sigma^2 t}-1)$. The log-return $\ln(S_t/S_0)\sim\mathcal{N}(\mu t,\sigma^2 t)$.
Practice
- Show $W_t^2-t$ is a martingale using properties of BM.
- Compute $E[W_s W_t]$ for $s\le t$ using independent increments.
- Explain why $[W]_t=t$ rules out differentiable paths.
Show Answer Key
1. By Itô's lemma, $d(W_t^2)=2W_t\,dW_t+dt$, so $d(W_t^2-t)=2W_t\,dW_t$. Since the $dt$ term vanishes, $W_t^2-t$ is a local martingale; bounded on compacts, hence a martingale.
2. $E[W_sW_t]=E[W_s(W_s+(W_t-W_s))]=E[W_s^2]+E[W_s]E[W_t-W_s]=s+0=s=\min(s,t)$.
3. The total quadratic variation $[W]_t=t>0$ is incompatible with differentiable paths, which would have $[W]_t=0$. Hence BM paths are nowhere differentiable (a.s.).