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### Ex4

Course: MS 334, Fall 2009
School: East Los Angeles College
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Word Count: 154

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334 MS Advanced Stochastic Modelling, Spring 2008 Exercise Sheet 4 1. Let A(t) be absorbed Brownian motion with A(0) = c where c &gt; 0 is fixed. (a) Show that Sn = A(n) is a martingale. (b) Let M = sup{A(n) : n 1}. Estimate P (M &gt; 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...

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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...

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