Brownian Motion

Standard Brownian motion $W(t)$ is the continuous-time limit of a symmetric random walk. It's defined by four properties:

$W(0) = 0$.Continuous paths.Independent increments: $W(t + s) - W(t)$ is independent of the past up to $t$.Gaussian increments: $W(t + s) - W(t) \sim N(0, s)$.

Several remarkable consequences. Brownian paths are continuous everywhere but differentiable nowhere — they're so jagged that ordinary calculus breaks down. The variance grows linearly with time, but the path itself wanders by $\sqrt{t}$ on typical scales.

Brownian motion is the foundation of continuous-time finance. Stock prices in the Black-Scholes world are functions of Brownian motion. Yield curve dynamics, exchange rates, and short-rate models all build on it. To do calculus on Brownian-driven processes, you need Itô calculus — a topic we'll get to next.