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

Related Topics:
Probability theory - Discrete-time - Stochastic process - Sequence - Random variable - That

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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.

Related Topics:
Expected value - 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

Related Topics:
Continuous-time - Zero-drift - Stochastic process - Random variable - 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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