*You may want to read this article first: What is conditional expectation?*

## What is a Martingale?

A **martingale** is model of a fair game. It is a sequence of random variables x_{0}, x_{1}, x_{2}…x_{n} with one important property: the conditional expectation of x_{n+1} given x_{0}, x_{1}, x_{2}…x_{n} is always just x_{n}.

In other words, it is a sequence of random variables such that for any time n:

E(|X_{n}|) < ∞

E(X_{n + 1}| X_{1}…, X_{n}) = X_{n}.

## History of Martingales

The word *martingale* came from a group of betting strategies that were popular in France in the 18th century. In a simple game where a gambler wins if a coin comes up heads and loses if it comes up tails (p_{t} = p_{h} = 1/2, assuming a fair coin) the martingale strategy had him double his bet every time he lost. As he continues to play, the probability of tossing at least one head approaches 1 (as the number of trials approach infinity), so he was considered to be certain of eventually winning back everything he lost plus his original stake.

Since the amount staked increases exponentially with this method and no player actually possesses the infinite bankrolls necessary to ensure success, it is rather more risky than one might think.

In probability theory, the concept of martingales was pioneered by Paul Levy in 1934. The term was first used for the statistical concept by Jean Ville in 1939.

## Examples of Martingales

A simple example of a martingale is a one-dimensional random walk, where steps are equally likely in either direction. Here, for each step, p_{left} = p_{right} = 1/2.

In fact, an unbiased random walk in any number of dimensions is a martingale.

Under the unified neutral theory of biodiversity and biogeography, the species count for any particular species of fixed size in an ecological community will be a function of discrete time. We can consider this as a sequence of random variables, and the sequence is a martingale.

## Variants on Martingales

A sequence such that

E(|X

_{n}|) < ∞

E(X_{n + 1}| X_{1}…, X_{n}) ≥ X_{n}.

Is called a submartingale.

Similarly, a sequence such that

E(|X

_{n}|) < ∞

E(X_{n + 1}| X_{1}…, X_{n}) ≤ X_{n}.

Is known as a supermartingale.

## References

Doob, J.L. What is a Martingale? The American Mathematical Monthly

Vol. 78, No. 5 (May, 1971), pp. 451-463. Retrieved from http://www.jstor.org/stable/2317751 on March 21, 2018

Williams, D. (1991). Probability with Martingales. Cambridge University Press.