Continuous-Time Models

Defining properties of standard Brownian motion Wₜ

(1) W₀ = 0; (2) continuous sample paths; (3) independent increments; (4) Wₜ − Wₛ ∼ N(0, t − s) for 0 ≤ s < t.

Mean and variance of Wₜ

𝔼[Wₜ] = 0 and Var(Wₜ) = t. Hence Wₜ ∼ N(0, t).

Covariance of Brownian motion

Cov(Wₛ, Wₜ) = min(s, t).

Quadratic variation of Brownian motion on [0, T]

⟨W⟩ₜ = t almost surely. Brownian motion has finite quadratic variation but unbounded total variation.

Itô multiplication table

dWₜ · dWₜ = dt, dWₜ · dt = 0, dt · dt = 0. For correlated BMs W, W̃ with correlation ρ: dWₜ · dW̃ₜ = ρ dt (and = 0 if