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CONDITIONAL EXPECTATION AND MARTINGALES

CONDITIONAL EXPECTATION AND MARTINGALES1. DI S CR E T E-TI MEMART I N GA L E of a {Fn}n 0be an increasing sequence of algebras in aprobability space ( ,F,P). Such a sequence will be called afiltration. LetX0,X1, .. be anadaptedsequence ofintegrablereal-valued random variables, that is, a sequence with the prop-erty that for eachnthe random variableXnis measurable relative toFnand such thatE|Xn|< . The sequenceX0,X1, .. is said to be amartingalerelative to the filtration {Fn}n 0if it isadapted and if for everyn,(1)E(Xn+1|Fn)= , it is said to be asupermartingale(respectively,submarting ale) if for everyn,(2)E(Xn+1|Fn) ( ) that any martingale is automatically both a submartingale and a and Martingale Difference most basic examples of martin-gales are sums of independent, mean zero random variables.

adapted sequence of integrable real-valued random variables, that is, a sequence with the prop-erty that for each n the random variable Xn is measurable relative to Fn and such that EjXnj˙ 1. The sequence X0,X1,... is said to be a martingale relative to the filtration {Fn}n‚0 if it is adapted and if for every n, (1) E(Xn¯1 jFn) ˘ Xn.

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