Exercise 13.3.5 Let { X(t), t 0 } be a (0, ) Brownian motion. Prove that the following three processes are martingales: (1) X(t),

Exercise 13.3.5 Let { X(t), t ≥ 0 } be a (0, σ) Brownian motion. Prove that the following three processes are martingales: (1) X(t), (2) X(t)2 −σ2t, and

(3) exp[ αX(t)−α2σ2t/2 ] for α ∈ R, called Wald’s martingale.