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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 < s < 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...
View Full Document

Ask a homework question - tutors are online