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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
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).
- Spring '10
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