Martingale (probability theory)


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.Image:HittingTimes1.png|thumb|340px|Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy.

History

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.

Definitions

A basic definition of a discrete-time martingale is a discrete-time stochastic process 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.

Martingale sequences with respect to another sequence

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, is equal to the observation at time s. The second property implies that is measurable with respect to.

General definition

In full generality, a stochastic process taking values in a Banach space with norm is a martingale with respect to a filtration 'and probability measure ' if
  • Σ∗ is a filtration of the underlying probability space ;
  • Y is adapted to the filtration Σ∗, i.e., for each t in the index set T, the random variable Yt is a Σt-measurable function;
  • for each t, Yt lies in the Lp space L1, i.e.
  • for all s and t with s < t and all F ∈ Σs,
It is important to note that the property of being a martingale involves both the filtration and the probability measure. It is possible that Y could be a martingale with respect to one measure but not another one; the Girsanov theorem offers a way to find a measure with respect to which an Itō process is a martingale.
In the Banach space setting the conditional expectation is also denoted in operator notation as.

Examples of martingales

  • An unbiased random walk, in any number of dimensions, is an example of a martingale. For example, consider a 1-dimensional random walk where at each time step a move to the right or left is equally likely.
  • A gambler's fortune is a martingale if all the betting games which the gambler plays are fair. The gambler is playing a game of coin flipping. Suppose Xn is the gambler's fortune after n tosses of a fair coin, such that the gambler wins $1 if the coin toss outcome is heads and loses $1 if the coin toss outcome is tails. The gambler's conditional expected fortune after the next game, given the history, is equal to his present fortune. This sequence is thus a martingale.
  • Let Yn = Xn2 − n where Xn is the gambler's fortune from the prior example. Then the sequence is a martingale. This can be used to show that the gambler's total gain or loss varies roughly between plus or minus the square root of the number of games of coin flipping played.
  • de Moivre's martingale: Suppose the coin toss outcomes are unfair, i.e., biased, with probability p of coming up heads and probability q = 1 − p of tails. Let
  • Pólya's urn contains a number of different-coloured marbles; at each iteration a marble is randomly selected from the urn and replaced with several more of that same colour. For any given colour, the fraction of marbles in the urn with that colour is a martingale. For example, if currently 95% of the marbles are red then, though the next iteration is more likely to add red marbles than another color, this bias is exactly balanced out by the fact that adding more red marbles alters the fraction much less significantly than adding the same number of non-red marbles would.
  • Likelihood-ratio testing in statistics: A random variable X is thought to be distributed according either to probability density f or to a different probability density g. A random sample X1,..., Xn is taken. Let Yn be the "likelihood ratio"
Image:Martingale1.svg|thumb|250px|Software-created martingale series
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 but instead an upper or lower bound on the conditional expectation. These generalizations reflect the relationship between martingale theory and potential theory, that is, the study of harmonic functions. Just as a continuous-time martingale satisfies E − Xs = 0 ∀s ≤ t, a harmonic function f satisfies the partial differential equation Δf = 0 where Δ is the Laplacian operator. Given a Brownian motion process Wt and a harmonic function f, the resulting process f is also a martingale.
  • A discrete-time submartingale is a sequence of integrable random variables satisfying
  • Analogously, a discrete-time supermartingale satisfies

    Examples of submartingales and supermartingales

  • Every martingale is also a submartingale and a supermartingale. Conversely, any stochastic process that is both a submartingale and a supermartingale is a martingale.
  • Consider again the gambler who wins $1 when a coin comes up heads and loses $1 when the coin comes up tails. Suppose now that the coin may be biased, so that it comes up heads with probability p.
  • * If p is equal to 1/2, the gambler on average neither wins nor loses money, and the gambler's fortune over time is a martingale.
  • * If p is less than 1/2, the gambler loses money on average, and the gambler's fortune over time is a supermartingale.
  • * If p is greater than 1/2, the gambler wins money on average, and the gambler's fortune over time is a submartingale.
  • A convex function of a martingale is a submartingale, by Jensen's inequality. For example, the square of the gambler's fortune in the fair coin game is a submartingale. Similarly, a concave function of a martingale is a supermartingale.

    Martingales and stopping times

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. The intuition behind the definition is that at any particular time t, you can look at the sequence so far and tell if it is time to stop. 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, 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.
One of the basic properties of martingales is that, if is a martingale and is a stopping time, then the corresponding stopped process defined by is also a martingale.
The concept of a stopped martingale leads to a series of important theorems, including, for example, the optional stopping theorem which states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.