1. Let B: be Brownian motion on R, Bo = 0. a) Prove that the moment generating function of B, satisfies E [(" Bi] = exp for all u e R. b) Use the moment generating function in a) to calculate E [ Bi] = 3t2 and more generally that E[ BZW] = (2k)! the . 2k . k! KEN c) The following is the statement of Kolmogorov's continuity theorem: Suppose that the process X = {X},, satisfies the following condition: For all 7 > 0 there exist positive constants o, B, D such that EllXt - X.["]