Problem 5 (30 points). Given a sample space of events ? = {H,T}, we consider the sequence of random i.i.d. variables (X});20 defined on ? = {H,T} by Xo =0 and Xi(wj) = +1, if wj = H for j 2 1. -1, if wj =T We define now a simple random walk (M,)nzo by Mo = 0 and M, = ) X, for any n 2 1. Throughout this problem, we use the notations p = P(H) and q = P(T) for a given probability measure P. (5.1) In this question, we assume that p = q = 1/2 in which case (Mn)n20 is called a Symmetric Random Walk. (a) Show that the sequence (Mn)n20 is a Martingale. (b) Show that the sequence (Zn)n20 defined by Zn = M2 - n is a Martingale. (c) Show that the sequence (Y,)n20 defined by Y, = M2 - 3nM, is a Martingale. (d) Show that the sequence (S,),20 defined by S, = (20per ) e"M. is a Martingale for o > 0. (5.2) We assume in this question that p = 1/3. (a) Is the sequence (Mn )n20 a Supermartingale or a Submartingale? (b) Is the sequence (Zn)n20 a Supermartingale or a Submartingale? (5.3) Prove that the sequence (Ln)n20 defined by In = (1," ) is a Martingale for p > 0. Problem 6 (10 points). Given the Symmetric Random Walk (M,),>o introduced in Problem 5, we define the sequence (In)n20 as lo = 0 and In = _ M;(Miti - M;) for n 2 1. j=0 (6.1) Show that the sequence (In )n20 is a Martingale. (6.2) Show that the sequence (In)n20 is a Markov Process