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Unformatted text preview: (XY |A) = XE (Y |A) 5.2 Examples, Basic Properties We begin by giving the definition of a martingale. Thinking of Mn as the amount of money at time n for a gambler betting on a fair game, and Xn as the outcomes of the gambling game we say that M0 , M1 , . . . is a martingale with respect to X0 , X1 , . . . if for any n 0 we have E |Mn | < 1 and for any possible values xn , . . . , x0 E (Mn+1 Mn |Xn = xn , Xn 1 = xn 1 , . . . X0 = x0 , M0 = m0 ) = 0 (5.4) The first condition, E |Mn | < 1, is needed to guarantee that the conditional expectation makes sense. The second, defining property, says that conditional on the past up to time n, the average profit from the bet on the nth game is 0. It will take several examples to explain why this is a useful definition. In many of our examples Xn will be a Markov chain and Mn = f (Xn , n). The conditioning event is formulated in terms of Xn because in passing from the random variables Xn that are driving the process the martingale to Mn , there 2 may be a loss of information...
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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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