Gambler's Ruin Problem. A gambler starts playing a two-outcome betting game, starting with an initial wealth x = R. Each time the gambler bets $1, he wins with probability p = (0, 1) and will stop when the total wealth reaches either 0 (he goes bankrupt) or a fixed amount M > x. Of course, the resulting process is a Random Walk with absorbing states at 0 or M. Let {X}n1 be the result of the n-th game. Then the sequence {X1, X2,...} is i.i.d., each taking values +1 or -1 with probability p and q = 1-p, respectively. The wealth at time n 0 is given by: n Wn=Xi, Xo = x. i=0 X=2. It is easy to see that the "profit" process Sn = Wn-x is an equivalent process, starting at 0 and stopping when reached either -x or M - x. a) Assume that the game is fair, i.e. p = 1/2. Show that Sn is a martingale. b) Argue that is a stopping time, where T = min{n 0: Sn = -x or Sn = M - x} c) Assume that the game is fair, and let po be the probability of ruin or, put it differently, the probability that Sn reaches -x before reaching M - x. Find po using the optional sampling theorem. d) Let us further calculate the expected time to ruin for a fair game. First, show that Zn = S a martingale. Then, apply the optional sampling theorem for Zn to obtain E[7]. e) Assume now that the game is not fair, i.e. p 1/2. In order to apply the same reasoning, we need to construct a martingale. Check if Sn is in this case a martingale, a supermartingale, or a submartingale. Prove that Sn Yn = is a martingale. Further, use the optimal sampling theorem to calculate the probability of ruin po.