Continuous-Time Models

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

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

Cov(Wₛ, Wₜ) = min(s, t).

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

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

For simple adapted Hₜ = Σ ξᵢ 1_(tᵢ, tᵢ₊₁], ∫₀ᵀ Hₜ dWₜ := Σ ξᵢ (W_(tᵢ₊₁) − W_(tᵢ)). Extend by L² isometry to all progressively measurable H with 𝔼∫₀ᵀ Hₜ² dt < ∞.