Stochastic

N j 2j 1 j 2n 1 n 1 a little thought shows

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Unformatted text preview: n m=n Using the assumptions ET < 1 and E |X | < 1 one can prove that the righthand side tends to 0 and complete the proof. However the details are somewhat complicated and are not enlightening so they are omitted. Our next two examples are applications of the exponential martingale in Example 5.6: Example 5.12. Left-continuous random walk. Suppose that X1 , X2 , . . . are independent integer-valued random variables with EXi > 0, P (Xi 1) = 1, and P (Xi = 1) > 0. These walks are called left-continuous since they cannot jump over any integers when they are decreasing, which is going to the left as the number line is usually drawn. Let (✓) = exp(✓Xi ) and define ↵ < 0 by the requirement that (↵) = 1. To see that such an ↵ exists, note that (i) (0) = 1 and 0 (✓) = d Ee✓xi = E (xi e✓xi ) d✓ so 0 (0) = Exi > 0 and it follows that (✓) < 1 for small negative ✓. (ii) If ✓ < 0, then (✓) e ✓ P (xi = 1) ! 1 as ✓ ! 1. Our choice of ↵ makes exp(↵Sn ) a martingale....
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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