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CONDITIONAL EXPECTATION AND MARTINGALES

CONDITIONAL EXPECTATION AND MARTINGALES1. IN T RO D U CT I O NMartingalesplay a role in stochastic processes roughly similar to that played byconservedquantitiesin dynamical systems. Unlike a conserved quantity in dynamics, which remainsconstant in time, a martingale s value can change; however, itsexpectationremains constantin time. More important, the EXPECTATION of a martingale is unaffected byoptional fact, this can be used as a provisional definition: A discrete-timemartingaleis a sequence{Xn}n 0of integrable real (or complex) random variables with the property that for every boundedstopping time , theOptional Sampling Formula(1)E X =E X0is have seen the Optional Sampling Formula before, in various guises.

conditional expectations behave like ordinary expectations, with random quantities that are functions of the conditioning random variable being treated as constants.2 Let Y be a random variable, vector, or object valued in a measurable space, and let X be an integrable random variable (that is, a random variable with EjXj˙1).

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