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A martingale is the mathematical description of a fair game: The expected net gain or loss from further play, independent of the history, is 0. The rigorous mathematical definition involves conditional expectation. Following earlier work by Paul Lévy and Jean Ville, Joseph Leo Doob developed in the 1940s and 1950s basic martingale theory and many of its applications. Martingales allowed one to study, for the first time, the behavior of sums and sequences of random variables which are not independent. Martingale theory is one of the cornerstones of modern mathematical probability theory with wide-ranging applications in stochastic analysis and mathematical finance.


Original languageEnglish
Title of host publicationInternational Encyclopedia of the Social & Behavioral Sciences: Second Edition
PublisherElsevier Inc.
Number of pages5
ISBN (electronic)9780080970875
ISBN (print)9780080970868
Publication statusPublished - 26 Mar 2015


ASJC Scopus subject areas


  • Banach-space geometry, Berry-Esseen theorem, Central limit theorem, Conditional probability, Fair game, Gambling strategy, Game theory, Harmonic analysis, Likelihood ratio, Martingale, Mathematical finance, Optimal stopping, Stochastic (Itô) integral, Stochastic analysis, Stochastic process