Martingale (probability theory)

In probability theory, a (discrete-time) martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X₁, X₂, X₃, ... that satisfies the identity

$E\left(X_{n+1}\mid X_{1},\dots ,X_{n}\right)=X_{n},$

i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the most recent past observation. Like many things in probability theory, the term was adopted from the language of gambling.

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

$E\left(Y_{n+1}\mid X_{1},\dots ,X_{n}\right)=Y_{n},$

for every n.

Examples of martingales

• Suppose X_n is a gambler's fortune after n tosses of a "fair" coin, where the gambler wins $1 if the coin comes up heads and loses $1 if the coin comes up tails. The gambler's conditional expected fortune after the next trial, given the history, is equal to his present fortune, so this sequence is a martingale.

• Let Y_n = X_n² - n where X_n is the gambler's fortune from the preceding example. Then the sequence { Y_n : n = 1, 2, 3, ... }is a martingale. This can be used to show that the gambler's total gain or loss grows roughly as the square root of the number of steps.

• (de Moivre's martingale) Now suppose an "unfair" or "biased" coin, with probability p of "heads" and probability q = 1 − p of "tails". Let

$X_{n+1}=X_{n}\pm 1$

with "+" in case of "heads" and "-" in case of "tails". Let

$Y_{n}=(q/p)^{X_{n}}.$

Then { Y_n : n = 1, 2, 3, ... } is a martingale with respect to { X_n : n = 1, 2, 3, ... }.

• Let Y_n = P(A | X₁, ... , X_n). Then { Y_n : n = 1, 2, 3, ... } is a martingale with respect to { X_n : n = 1, 2, 3, ... }.

• (Polya's urn) An urn initially contains r red and b blue marbles. One is chosen randomly. If it is red, it is replaced and a new red marble put into the urn. If it is blue, it is replaced and a new blue marble put into the urn. Let X_n be the number of red marbles in the urn after n iterations of this procedure, and let Y_n=X_n/(n+r+b). Then the sequence { Y_n : n = 1, 2, 3, ... } is a martingale.

• (Likelihood-ratio testing in statistics) A population is thought to be distributed according either to a probability density f or another probability density g. A random sample is taken, the data being X₁, ... , X_n. Let Y_n be the "likelihood ratio"

$Y_{n}=\prod _{i=1}^{n}{\frac {g(X_{i})}{f(X_{i})}}$

(which, in applications, would be used as a test statistic). Then { Y_n : n = 1, 2, 3, ... } is a martingale with respect to { X_n : n = 1, 2, 3, ... }.

• Suppose each ameba either splits into two amebas, with probability p, or eventually dies, with probability 1 - p. Let X_n be the number of amebas surviving in the nth generation (in particular X_n = 0 if the population has become extinct by that time). Let r be the probability of eventual extinction. (Finding r as function of p is an instructive exercise. Hint: The probability that the descendants of an ameba eventually die out is equal to the probability that either of its immediate offspring dies out, given that the original ameba has split.) Then

$\left\{\,r^{X_{n}}:n=1,2,3,\dots \,\right\}$

is a martingale with respect to { X_n: n = 1, 2, 3, ... }.

Convergence of martingales

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Martingales and stopping times

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Submartingales and supermartingales

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