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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 EXPECTATION AND MARTINGALES 1. INTRODUCTION Martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. Unlike a conserved quantity in dynamics, which remains constant in time, a martingale’s value can change; however, its expectation remains constant in time.

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