Let W = {Wt : t 0} be a Brownian motion on F = P) I Show that W is an F-martingale. 2 Show that for every a R. the process xta exp(aWt *02t) is an F-martingale. (20% of total points) (20% of total points) Hint: The MGF of X NO", "2) is Mx(a) = exp(ga -+ 3 Dehne the polynomials Hn(x, y); n = O, I, 2, by Hn(x, y) = exp(ax iQ2y) at a = 0. For example, It can be shown (using Taylor series) that Xta = exp(aVVt -102 t) = = 3xy, y) 6x2y+3y2, etc. Hn(Wt, t). We now show that Hn(Wt, t) is a martingale for each n. (a) Let O s t and a R Explain why for each F (10* of total points) (b) By differentiating (3a) on both sides n times with respect to and interchanging the derivative with the integral (no need to justify this step), show (c) Conclude that : t 0} is a martingale. (10% of total points) (10% of total points)