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Unformatted text preview: r the six state chain defined in Exercise 1.66. Show that the total number of A’s is a martingale and use this to compute the probability of getting absorbed into the 2,2 (i.e., all A’s state) starting from each initial state. 5.2. Let Xn be the Wright–Fisher model with no mutation defined in Example 1.9. (a) Show that Xn is a martingale and use Theorem 5.14 to conclude that Px (VN < V0 ) = x/N . (b) Show that Yn = Xn (N Xn )/(1 1/N )n is a martingale. (c) Use this to conclude that (N 1) x(N x)(1 1/N )n N2 Px (0 < Xn < N ) 4 5.3. Lognormal stock prices. Consider the special case of Example 5.5 in which Xi = e⌘i where ⌘i = normal(µ, 2 ). For what values of µ and is Mn = M0 · X1 · · · Xn a martingale? 5.4. Suppose that in Polya’s urn there is one ball of each color at time 0. Let Xn be the fraction of red balls at time n. Use Theorem 5.13 to conclude that P (Xn 0.9 for some n) 5/9. 5.5. Suppose that in Polya’s urn there are r red balls and g green balls at time...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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