Brownian Motion (Wiener Process)

Brownian motion (or the Wiener process) $W_t$ is the continuous-time limit of a random walk. It is the fundamental building block of stochastic calculus and mathematical finance. Defining properties: 1. $W_0 = 0$ 2. Independent increments: $W_t - W_s$ is independent of $W_u$ for all $u \leq s < t$ 3. Gaussian increments: $W_t - W_s \sim N(0, t-s)$ 4. Continuous paths: $t \mapsto W_t$ is continuous (but nowhere differentiable!) Intuition: Imagine flipping a coin every $\Delta t$ seconds, stepping $+\sqrt{\Delta t}$ or $-\sqrt{\Delta t}$. As $\Delta t \to 0$, this random walk converges to Brownian motion. The $\sqrt{\Delta t}$ scaling is crucial — it gives variance proportional to time. Key facts: - $E[W_t] = 0$ and $\text{Var}(W_t) = t$ - $E[W_s W_t] = \min(s,t)$ - The paths are continuous but have infinite variation (you can't define a classical integral) - $W_t / \sqrt{t} \sim N(0,1)$ for any $t > 0$ Concrete example: A stock price changes every second by a random amount. Over 1 day ($t=1$), the cumulative random component has standard deviation $\sigma$. Over 4 days, the standard deviation is $2\sigma$ (not $4\sigma$) — the square root of time scaling. When to use: Modeling random continuous phenomena — stock prices, particle diffusion, interest rates. It's the noise term in virtually every continuous-time financial model. Understanding Brownian motion is prerequisite to Ito's lemma and the Black-Scholes formula. This is a fundamental technique — the foundational stochastic process in quantitative finance.