334 MS Advanced Stochastic Modelling, Spring 2008 Exercise Sheet 4
1.
Let A(t) be absorbed Brownian motion with A(0) = c where c > 0 is fixed.
(a) Show that Sn = A(n) is a martingale. (b) Let M = sup{A(n) : n 1}. Estimate P (M > x|A(0) = c) for x (0, c). 2. The amount of money owned by a gambler playing a fair game is denoted by Sn , where S0 = 10. 3 If steps n-1 and n both result in wins, then the gambler bets 4 Sn on the (n+1)'th step 7 (so Sn+1 = 4 Sn or Sn+1 = 1 Sn with equal probability). If steps n - 1 and n 4 1 both result in losses, then the gambler bets 4 Sn on the (n + 1)'th step. Otherwise (including steps 1 and 2), the gambler bets 1 Sn on the (n + 1)'th step. 2 (a) Show (using tr...