Sunday, May 1, 2011

What is Martingale Probability Theory?

The Martingale technical definition.  (Definition from WikipediA with link to WikipediA)
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 martingale is a model of a fair game. Precise definitions are given below. Originally,
The concept of martingale in probability theory was introduced by
martingale referred to a class of betting strategies that was popular in 18th century France.
[1] The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, his probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users.Paul Pierre Lévy, and much of the original development of the theory was done by Joseph Leo Doob among others. Part of the motivation for that work was to show the impossibility of successful betting strategies.


Siminelakis, Paris (2010).
"The Splendors and Miseries of Martingales". Electronic Journal for History of Probability and Statistics 5 (1). June 2009. Entire issue dedicated to Martingale probability theory. Williams, David (1991). Probability with Martingales. Cambridge University Press. ISBN 0-521-40605-6. Kleinert, Hagen (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (4th ed.). Singapore: World Scientific. ISBN 981-238-107-4. "Martingales and Stopping Times: Use of martingales in obtaining bounds and analyzing algorithms" (PDF). University of Athens.

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