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Martingale is a stochastic process which has the following property: the expectation of any future value equals the value today. In other words, stochastic process X(t) is a martingale if, for any moments of time s < t,

                                                       E[X(t) | X(s)] = E[X(s)].                                           (M)

Martingale formalizes the idea of a fair game. Suppose we toss a fair coin many times. Each time we receive $1 if the coin lands heads and pay $1 if the coin lands tails. Then our cumulative wealth at any moment of time is a martingale.

Martingales can be defined in both discrete and continuous time. The way we played with a coin toss is an example of discrete time martingale dynamics. A major instance in continuous time is Brownian motion. Martingales fall at the boundary of super-martingales:

E[X(t) | X(s)] <= E[X(s)].

and sub-martingales:

E[X(t) | X(s)] >= E[X(s)].

A massive body of results has been derived for all three types of processes. In particular, the Optional Stopping Theorem extends the martingale property (M) for the case when t is random, under certain regularity conditions. This theorem enables one to use martingales for calculating exit probabilities of various processes out of predefined regions. Such problems have relevance in physics, actuarial science and risk management.


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