Martingales

A martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings. In particular, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.

To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time. However, knowledge of the prior outcomes (e.g., all prior cards drawn from a card deck) may be able to reduce the uncertainty of future outcomes. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning strategy is used. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games.

See also

Wiener process, Markov Process

Papers

• Siminelakis, Paris (2010). Martingales and Stopping Times: Use of martingales in obtaining bounds and analyzing algorithms. University of Athens.
• Brown, B. M. (1971). Martingale central limit theorems. The Annals of Mathematical Statistics, 42(1), 59-66.
• Mandelbrot, B. (1966). Forecasts of future prices, unbiased markets, and” martingale” models. The Journal of Business, 39(1), 242-255.

Books

• Hall, P., & Heyde, C. C. (2014). Martingale limit theory and its application. Academic press.
• Musiela, M., & Rutkowski, M. (2006). Martingale methods in financial modelling (Vol. 36). Springer Science & Business Media.