In probability theory, a martingale is a stochastic process in which the expected value of the next observation, given all prior observations, is equal to the most recent value. In other words, the conditional expectation of the next value, given the past, is equal to the present value. Martingales are used to model fair games, where future expected winnings are equal to the current amount regardless of past outcomes.
History Originally, martingale referred to a class of betting strategies that was popular in 18th-century France. The historical development of the concept can be summarized as follows:
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 the martingale in probability theory was introduced by Paul Lévy in 1934, though he did not formally name it. The term "martingale" was introduced later by Ville (1939), 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 early work was to mathematically prove the impossibility of successful betting strategies in games of chance.
Definitions A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } that satisfies for any time n {\displaystyle n} :
E [ | X n | ] < ∞ {\displaystyle \mathbf {E} [\vert X_{n}\vert ]<\infty }
E [ X n + 1 ∣ X 1 , … , X n ] = X n . {\displaystyle \mathbf {E} [X_{n+1}\mid X_{1},\ldots ,X_{n}]=X_{n}.}
That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.
Martingale sequences with respect to another sequence More generally, a sequence Y 1 , Y 2 , Y 3 , … {\displaystyle Y_{1},Y_{2},Y_{3},\dots } is said to be a martingale with respect to another sequence X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } if for all n {\displaystyle n} :
E [ | Y n | ] < ∞ {\displaystyle \mathbf {E} [\vert Y_{n}\vert ]<\infty }
E [ Y n + 1 ∣ X 1 , … , X n ] = Y n . {\displaystyle \mathbf {E} [Y_{n+1}\mid X_{1},\ldots ,X_{n}]=Y_{n}.}
Similarly, a continuous-time martingale with respect to the stochastic process X t {\displaystyle X_{t}} is a stochastic process Y t {\displaystyle Y_{t}} such that for all t {\displaystyle t} :
E [ | Y t | ] < ∞ {\displaystyle \mathbf {E} [\vert Y_{t}\vert ]<\infty }
E [ Y t ∣ { X τ , τ ≤ s } ] = Y s ∀ s ≤ t . {\displaystyle \mathbf {E} [Y_{t}\mid \{X_{\tau },\tau \leq s\}]=Y_{s}\quad \forall s\leq t.}
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