Given a sample space of events = {H, T}, we consider the sequence of random i.i.d. variables (Xn)n0 defined on = {H, T} by X0 = 0 and Xj (j ) = +1, if j = H 1, if j = T for j 1. We define now a simple random walk (Mn)n0 by M0 = 0 and Mn = Xn j=1 Xj for any n 1.

(1) We assume that P(H) = P(T) = 1/2. (a) Show that (Mn)n0 is a Martingale. (b) Show that the sequence (Zn)n0 defined by Zn = M2 n n is a Martingale.

Problem 5 (20 points). Given a sample space of events ? = {H, T}, we consider the sequence of random i.i.d. variables (Xn),30 defined on ? = {H. T} by Xo = 0 and Xi(wy) = +1, if wj = H -1, if wj = T for j 2 1. FE We define now a simple random walk (M,),>o by Mo =0 and M, = EX; for any n 2 1. j= 1 (1) We assume that P(H ) = P(T) = 1/2. (a) Show that (My),20 is a Martingale. (b) Show that the sequence (Z,),20 defined by Z, = M. -n is a Martingale. (2) We assume now that P(H) = 2/3. (a) Is (Mm)n20 a Supermartingale or a Submartingale? (b) Is (Zn ),20 a Supermartingale or a Submartingale