Stochastic process
Martingale

A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general.

A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. For a sequence of independent and identically distributed random variables X 1, X 2, X 3, … {\displaystyle X_{1},X_{2},X_{3},\dots } with zero mean, the stochastic process formed from the successive partial sums X 1, X 1 + X 2, X 1 + X 2 + X 3, … {\displaystyle X_{1},X_{1}+X_{2},X_{1}+X_{2}+X_{3},\dots } is a discrete-time martingale. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process. Martingales can also be built from other martingales. For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.

Martingales mathematically formalize the idea of a 'fair game' where it is possible form reasonable expectations for payoffs, and they were originally developed to show that it is not possible to gain an 'unfair' advantage in such a game. But now they are used in many areas of probability, which is one of the main reasons for studying them. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.

Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance.
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