# Let aij be the prot for the ith security when the j

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Unformatted text preview: .3 we showed that if V0 = min{n 0 : Sn = 0} then Ex V0 = x/(1 2p). The aim of this problem is to compute the variance of V0 . (a) Show that (Sn (p q )n)2 n(1 (p q )2 ) is a martingale. (b) Use this to conclude that when S0 = x the variance of V0 is x· 1 (p q )2 (p q )3 (c) Why must the answer in (b) be of the form cx? 5.12. Generating function of the time of gambler’s ruin. Continue with the set-up of the previous problem. (a) Use the exponential martingale and our stopping theorem to conclude that if ✓ 0, then e✓x = Ex ( (✓) V0 ). (b) Let 0 &lt; s &lt; 1. Solve the equation (✓) = 1/s, then use (a) to conclude !x p 1 1 4pqs2 V0 Ex (s ) = 2ps (c) Why must the answer in (b) be of the form f (s)x ? 5.13. Consider a favorable game in which the payo↵s are 1, 1, or 2 with probability 1/3 each. Use the results of Example 5.12 to compute the probability we ever go broke (i.e, our winnings Wn reach \$0) when we start with \$i. 177 5.6. EXERCISES 5.14. A branching process can be turned into a random walk i...
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