Martingle

History

Originally, martingale referred to a class of betting strategies popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since as a gambler's wealth and available time jointly approach infinity his probability of eventually flipping heads approaches 1, the martingale betting strategy was seen as a sure thing by those who practiced it. Of course in reality the exponential growth of the bets would eventually bankrupt those foolish enough to use the martingale for a long time.

The concept of martingale in probability theory was introduced by Paul Pierre Lévy, and much of the original development of the theory was done by Joseph Leo Doob. Part of the motivation for that work was to show the impossibility of successful betting strategies.

Definitions

A discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies for all n

E(|Xn|) < infinity

E(Xn+1|X1,...,Xn)= Xn,

i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the last observation.

Somewhat more generally, a sequence Y1, Y2, Y3, ... is said to be a martingale with respect to another sequence X1, X2, X3, ... if for all n

E(|Yn|)< infinity

E(Yn+1|X1,.....,Xn) = Yn.

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