TABLE OF CONTENTS Stochastic processes as martingales have extensive applications in stochastic problems. They arise naturally whenever one needs to consider mathematical expectations with respect to increasing information patterns. They are used to state several theoretical results concerning the convergence and the convergence rate of learning systems (derivation of the asymptotic properties of recursive algorithms). The martingale, supermartingale and submartingale processes are defined in what follows.
A stochastic process { x n} is a martingale if it is uniformly integrable
and for any n = 1, 2, ...
A stochastic process { x n} is supermartingale if it is uniformly integrable and
A stochastic process { x n} is a submartingale if it is uniformly integrable and
The name " martingale" derives from a French acronym for the gambling strategy of doubling one's bets until a win is secured. Let x n be the player's fortune at stage n of a game. The martingale property captures one notion of a game being fair in that the player's fortune on the next play is, on average, his current fortune and is not otherwise affected by the previous history.
that { x n} is a supermatingale with respect to F n if and only if { ? x n} is a submartingale. Similarly, { x n} is a martingale with...
