TOPICS
Search

Martingale


A sequence of random variates X_0, X_1, ... with finite means such that the conditional expectation of X_(n+1) given X_0, X_1, X_2, ..., X_n is equal to X_n, i.e.,

 <X_(n+1)|X_0,...,X_n>=X_n

(Feller 1971, p. 210). The term was first used to describe a type of wagering in which the bet is doubled or halved after a loss or win, respectively. The concept of martingales is due to Lévy, and it was developed extensively by Doob.

A one-dimensional random walk with steps equally likely in either direction (p=q=1/2) is an example of a martingale.


See also

Absolutely Fair, Gambler's Ruin, Random Walk--1-Dimensional, Saint Petersburg Paradox

Explore with Wolfram|Alpha

References

Doob, J. L. Stochastic Processes. New York: Wiley, 1953.Feller, W. "Martingales." §6.12 in An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 210-215, 1971.Lévy, P. Calcul de probabilités. Paris: Gauthier-Villars, 1925.Lévy, P. Théorie de l'addition des variables aléatoires. Paris: Gauthier-Villars, 1954.Lévy, P. Processus stochastiques et mouvement Brownien, 2nd ed. Paris: Gauthier-Villars, 1965.Loève, M. Probability Theory I, 4th ed. New York: Springer-Verlag, 1977.

Referenced on Wolfram|Alpha

Martingale

Cite this as:

Weisstein, Eric W. "Martingale." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Martingale.html

Subject classifications