Random Walks & Brownian Motion
Discrete walks rescaled into Brownian motion; properties of Wiener paths.
A simple symmetric random walk on $\mathbb Z$ takes $\pm 1$ steps with probability $1/2$ each. After $n$ steps,
$$S_n = \sum_{i=1}^n X_i, \quad \mathbb E[S_n] = 0, \quad \mathrm{Var}(S_n) = n.$$By the central limit theorem $S_n/\sqrt n$ converges to $\mathcal N(0,1)$. Rescale space and time by $1/\sqrt n$ and $1/n$ respectively; the limiting process is Brownian motion $B(t)$, a continuous Gaussian process with:
- $B(0) = 0$, almost surely continuous.
- Independent increments: $B(t) - B(s) \sim \mathcal N(0, t-s)$ for $s < t$.
- Quadratic variation $[B,B](t) = t$ — the path is nowhere differentiable but has finite total $L^2$ "speed."
The Wiener measure on $C([0,T])$ is concentrated on continuous paths. Itô calculus extends ordinary calculus to functions of Brownian motion, with the celebrated identity $d(B_t^2) = 2 B_t\, dB_t + dt$ — extra "$dt$" comes from non-zero quadratic variation.