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Concept
Martingale (probability theory)
Summary
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.
Originally, martingale referred to a class of betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their 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, their 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 due to finite bankrolls. Stopped Brownian motion, which is a martingale process, can be used to model the trajectory of such games.
The concept of martingale in probability theory was introduced by Paul Lévy in 1934, though he did not name it. The term "martingale" was introduced later by , who also extended the definition to continuous martingales. 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 in games of chance.
A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies for any time n,
That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.
More generally, a sequence Y1, Y2, Y3 ... is said to be a martingale with respect to another sequence X1, X2, X3 ...
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Related concepts (27)
Stopping time
In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time) is a specific type of “random time”: a random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest. A stopping time is often defined by a stopping rule, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time.
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown.
Martingale (probability theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Originally, martingale referred to a class of betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails.
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Introduction to the mathematical theory of stochastic calculus: construction of stochastic Ito integral, proof of Ito formula, introduction to stochastic differential equations, Girsanov theorem and F