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ch2sum

# ch2sum - Chapter 2 Summary Denition We say that M0 M1 is a...

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Chapter 2 Summary Definition. We say that M 0 , M 1 , . . . is a martingale with respect to X 0 , X 1 , . . . if for any n 0 (i) E | M n | < , (ii) M n can be determined from the values of M 0 and X m , m n , and (iii) for any possible values x n , . . . , x 0 E ( M n +1 - M n | X n = x n , X n - 1 = x n - 1 , . . . X 0 = x 0 ) = 0 Theorem 1. If M n is martingale then EM n = EM 0 . 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 0 , X 1 , . . . , X n . If, for example, T = min { n 0 : X n A } this is true since { T = n } = { X 0 A, . . . X n - 1 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. Theorem 3. Suppose M n is a martingale and let T is a stopping time (both with respect to X n ). If P ( T < ) = 1 and there is a constant K so that | M T n | ≤ K for all n , then EM T = EM 0 Review Problems 1. Let
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