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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).
- Spring '10
- The Land