Martingale (probability theory)
Originally, martingale referred to a class of betting strategies that was popular in 18th-century France.
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.
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.
More generally, a sequence Y1, Y2, Y3 ... is said to be a martingale with respect to another sequence X1, X2, X3 ... if for all n Similarly, a continuous-time martingale with respect to the stochastic process Xt is a stochastic process Yt such that for all t This expresses the property that the conditional expectation of an observation at time t, given all the observations up to time
if It is important to note that the property of being a martingale involves both the filtration and the probability measure (with respect to which the expectations are taken).
In the Banach space setting the conditional expectation is also denoted in operator notation as
[4] There are two generalizations of a martingale that also include cases when the current observation Xn is not necessarily equal to the future conditional expectation E[Xn+1 | X1,...,Xn] but instead an upper or lower bound on the conditional expectation.
Just as a continuous-time martingale satisfies E[Xt | {Xτ : τ ≤ s}] − Xs = 0 ∀s ≤ t, a harmonic function f satisfies the partial differential equation Δf = 0 where Δ is the Laplacian operator.
A stopping time with respect to a sequence of random variables X1, X2, X3, ... is a random variable τ with the property that for each t, the occurrence or non-occurrence of the event τ = t depends only on the values of X1, X2, X3, ..., Xt.
An example in real life might be the time at which a gambler leaves the gambling table, which might be a function of their previous winnings (for example, he might leave only when he goes broke), but he can't choose to go or stay based on the outcome of games that haven't been played yet.
In some contexts the concept of stopping time is defined by requiring only that the occurrence or non-occurrence of the event τ = t is probabilistically independent of Xt + 1, Xt + 2, ... but not that it is completely determined by the history of the process up to time t. That is a weaker condition than the one appearing in the paragraph above, but is strong enough to serve in some of the proofs in which stopping times are used.
