A *martingale* is a stochastic process which stays the same, on average. That is, the expected future value conditional on the present is equal to the current value. Examples include the wealth of a gambler as a function of time, assuming that he is playing a fair game. The canonical example of a continuous time martingale is Brownian motion and, in discrete time, a symmetric random walk is a martingale. As always, we work with respect to a filtered probability space . A process is said to be *integrable* if the random variables are integrable, so that .

**Definition 1** * A martingale, **, is an integrable process satisfying *

* for all **. *

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