Martingale

A sequence of random variates , , ... with finite means such that the conditional expectation of  given , , , ...,  is equal to , i.e.,

(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 () is an example of a martingale.

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.