Brownian motion — properties and path behaviour

Brownian motion is the canonical continuous-time stochastic process. It is the limit of random walks, the only continuous Lévy process with independent stationary increments, and the foundational object on which the entire continuous-time finance edifice is built.

Definition

A stochastic process W = {W(t) : t ≥ 0} is a (standard) Brownian motion if:

1. W(0) = 0.
2. Independent increments: W(t) - W(s) is independent of F_s for any s < t.
3. Stationary Gaussian increments: W(t) - W(s) ~ N(0, t - s).
4. Continuous paths: t ↦ W(t) is continuous (with probability 1).

Four facts you must own

1. E[W(t)] = 0, Var(W(t)) = t — the mean is zero, the variance grows linearly with time.
2. Cov(W(s), W(t)) = min(s, t) — older Brownian motions correlate less with later ones.
3. Paths are continuous but nowhere differentiable. The 'velocity' of Brownian motion is undefined.
4. Quadratic variation: Σ (W(t_i+1) - W(t_i))² → T as the partition refines. The 'sum of squared changes' converges to T even though the sum of absolute changes is infinite.

Why non-differentiability matters

If you tried to use a normal calculus differential dW/dt, it would be infinite (or undefined) everywhere. This is why we cannot just use ordinary calculus on Brownian motion — and why Itô's calculus (Modules 3-4) has to be developed from scratch with a fundamentally different chain rule.

Quadratic variation — the deep fact

lim_{||Π||→0} Σ_i (W(t_{i+1}) - W(t_i))²  =  T   (almost surely)

Where Π = {0 = t_0 < t_1 < ... < t_n = T} is a partition. The squared increments of BM accumulate at rate dt per unit time. In differential notation: (dW)² = dt. This is the keystone identity that drives Itô's lemma and all the calculus that follows.

Martingale property

E[W(t) | F_s] = W(s) for s ≤ t. Brownian motion has constant expected future value — it is a martingale. So is W(t)² - t (the variance correction makes it a martingale). The martingale framework is essential for risk-neutral pricing (Module 9).

Strong vs weak Markov

Markov: the future depends only on the present. Strong Markov: this also holds for stopping times, not just deterministic times. Brownian motion is strong Markov — useful for pricing American options where exercise is determined by a stopping time.

Variants

• BM with drift: X(t) = μt + σ W(t). Used for log prices.
• Geometric BM: S(t) = S(0) exp((μ - σ²/2)t + σ W(t)). Used for stock prices.
• BM with reflection: bounce off a barrier; used for FX inside a band.
• Brownian bridge: BM conditioned to return to a fixed value at time T.
• Multi-dimensional BM: vector W(t), components independent (or correlated via L Cholesky).

Computational facts

• Simulation: W(t + Δt) = W(t) + √Δt · Z, Z ~ N(0, 1). Trivial.
• Reflection principle: P(max_{s ≤ t} W(s) ≥ a) = 2 P(W(t) ≥ a). Closed-form barrier-option pricing.
• Lévy's characterisation: a continuous process with W(0) = 0 and W²(t) - t a martingale must be BM (in 1D).