In the following question Bt denotes a Brownian motion process with Bo = 0. 1. Let Xn, n = 0,1,2,... denote a Random Walk process, with Xo = 0 and Xn+1 Xn + Yn+1, where (Yn) is a sequence of i.i.d. random variables with mean = 1 and variance o2 = 1. (a) Let Mn = x2 - 2nX, + n(n-1). Show that Mn, n = 0,1,2,... is a martingale. (b) Show that Mn+1 - Mn and M, are uncorrelated. (c) Suppose now that Yi has exp(1) distribution, with the moment generating function m(u) = EeuY: = 1/(1 u), defined for u 0. Give the values of c so that the probability of ruin is exponentially small in 1. Justify your answer. You may use the following without proof, but you get 3 bonus points for a correct proof using calculus: the equation (1 - 1)ect = 1 has a unique positive solution for c > 1, and no positive solutions for c 0. Give the values of c so that the probability of ruin is exponentially small in 1. Justify your answer. You may use the following without proof, but you get 3 bonus points for a correct proof using calculus: the equation (1 - 1)ect = 1 has a unique positive solution for c > 1, and no positive solutions for c