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. In particular, theWald Identities I,II, and III are all instances of (1). Let 0, 1, .. be independent, identicallydistributed random variables, and letSn= 1+ 2+ nbe thenth partial sum. Denote by , 2, and ( ) the mean, variance, and moment generating function of 1, that is, =E 1, 2=E( 1 )2,and ( )=Eexp{ 1}.

For random variables defined on discrete proba-bility spaces, conditional expectation can be defined in an elementary manner: In particular, the conditional expectation of a discrete random variable X given the value y of another dis-crete random variable Y may be defined by (5) E(X jY ˘ y) ˘ X x xP(X ˘x jY ˘ y),

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  Variable, Space, Expectations, Random, Conditional, Random variables, Martingales, Conditional expectation and martingales

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