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Unformatted text preview: Chapter 2 Summary Definition. We say that M ,M 1 ,... is a martingale with respect to X ,X 1 ,... if for any n (i) E  M n  < , (ii) M n can be determined from the values of M and X m , m n , and (iii) for any possible values x n ,...,x E ( M n +1 M n  X n = x n ,X n 1 = x n 1 ,...X = x ) = 0 Theorem 1. If M n is martingale then EM n = EM . We say that T is a stopping time with respect to X n if the occurrence (or nonoccurrence) of the event we stop at time n can be determined by looking at the values of the process up to that time: X ,X 1 ,...,X n . If, for example, T = min { n 0 : X n A } this is true since { T = n } = { X 6 A,...X n 1 6 A,X n A } Recall that a b = min { a,b } . Using this notation M T n is the process stopped at time T . Theorem 2. If M n is a martingale and T is a stopping time (both with respect to X n ) then M T n is a martingale....
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This note was uploaded on 09/29/2011 for the course MATH 4740 taught by Professor Durrett during the Spring '10 term at Cornell University (Engineering School).
 Spring '10
 DURRETT

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