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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.

Lawler, G. F. (1995). Introduction to Stochastic Processes. New York: Chapman and Hall/CRC.

Ross, S. M. (1995). Stochastic Processes (2nd ed). New York: Wiley.

Karlin, S., & Taylor, H. M. (1975). A First Course in Stochastic Processes (2nd ed). New York: Academic Press.

Karatzas, I., & Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (2nd ed). New York: Springer.

Oksendal, B. K. (2002). Stochastic Differential Equations: An Introduction with Applications (5th ed). Springer-Verlag Berlin Heidelberg.

Durrett, R. (2004). Probability: Theory and Examples (3rd ed). Belmont, CA: Duxbury Press.

Williams, D. (1991). Probability with Martingales. Cambridge University Press.

Protter, P. E. (2005). Stochastic Integration and Differential Equations (2nd ed). Springer-Verlag Berlin Heidelberg.

Doob, J. L. (1953). Stochastic Processes. New York: John Wiley and Sons.

Gikhman, I. I., & Skorokhod, A. V. (2007). The Theory of Stochastic Processes III. Springer-Verlag Berlin Heidelberg.

Bremaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer.

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