Martingale

:A separate article treats the device for fastening horses' bridles or dogs' collars called a martingale. See martingale (fastener).

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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

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:E(X_{n+1}mid X_1,dots,X_n)=X_n,

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i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the last observation. As is frequent in probability theory, the term was adopted from the language of gambling.

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

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:E(Y_{n+1}mid X_1,dots,X_n)=Y_n, for every n.

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A continuous-time martingale is a zero-drift stochastic process. That is, a random variable heta follows a continuous-time martingale iff

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:d heta = sigma , dz

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where dz is a Wiener process and the variable sigma is a constant or a stochastic process that may depend on heta or other stochastic variables.

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Probability theory: Probability theory is the mathematical study of probability....

Discrete-time: Discrete time is non-continuous time. Sampling at non-continuous times results in discrete-time samples. For example, a newspaper may report the price of crude oil once every 24 hours. In general, the sampling period in discrete-time systems is constant, but in some cases non-uniform sampling is a...

Stochastic process: In the mathematics of probability, a stochastic process is a random function. In the most common applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series in applications) or a region of space (a stochastic process bein...