Prob Expert: Martingale

In probability theory, a martingale is a stochastic process (i.e., a sequence of random variables) such that the conditional expected value of an observation at some time t, given all the observations up to some earlier time s, is equal to the observation at that earlier time s.

A discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) $X_1, X_2, X_3, \ldots$ that satisfies for all n

$E( \vert X_n \vert )< \infty$

$E(X_{n+1}\mid X_1,\ldots,X_n)=X_n,$

i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the last observation.

Somewhat more generally, a sequence $Y_1, Y_2, Y_3, \ldots$ is said to be a martingale with respect to another sequence $X_1, X_2, X_3, \ldots$ if for all n