Martingale

In probability theory, a (discrete-time) martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X₁, X₂, X₃, ... that satisfies the identity

i.e., the conditional expected value of the next observation, given all of the past observations, is equal to the most recent past observation. Like many things in probability theory, the term was adopted from the language of gambling.

Somewhat more generally, a sequence Y₁, Y₂, Y₃, ... is said to be a martingale with respect to another sequence X₁, X₂, X₃, ... if

for every n.

Table of contents

1 History 2 Examples of martingales 3 Convergence of martingales 4 Martingales and stopping times 5 Submartingales and supermartingales 5.1 Examples of submartingales and supermartingales

History

Originally, martingale referred to a class of betting strategies popular in 18th century France. 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. Since a gambler with infinite wealth is guaranteed to eventually flip heads, the martingale betting strategy was seen as a sure thing by those who practiced it. Unfortunately, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would quickly bankrupt those foolish enough to use the martingale after even a moderately long run of bad luck.

Martingales in the probability-theory sense were invented by Lévy, and much of the original development of the theory was done by Doob. Part of the motivation for that work was to show the impossibility of successful betting strategies.

Examples of martingales

• Suppose X_n is a gambler's fortune after n tosses of a "fair" coin, where the gambler wins $1 if the coin comes up heads and loses $1 if the coin comes up tails. The gambler's conditional expected fortune after the next trial, given the history, is equal to his present fortune, so this sequence is a martingale.

• Let Y_n = X_n² − n where X_n is the gambler's fortune from the preceding example. Then the sequence { Y_n : n = 1, 2, 3, ... } is a martingale. This can be used to show that the gambler's total gain or loss grows roughly as the square root of the number of steps.

• (de Moivre's martingale) Now suppose an "unfair" or "biased" coin, with probability p of "heads" and probability q = 1 − p of "tails". Let

with "+" in case of "heads" and "-" in case of "tails". Let

Then { Y_n : n = 1, 2, 3, ... } is a martingale with respect to { X_n : n = 1, 2, 3, ... }.

• Let Y_n = P(A | X₁, ... , X_n). Then { Y_n : n = 1, 2, 3, ... } is a martingale with respect to { X_n : n = 1, 2, 3, ... }.

• (Polya's urn) An urn initially contains r red and b blue marbles. One is chosen randomly. If it is red, it is replaced and a new red marble put into the urn. If it is blue, it is replaced and a new blue marble put into the urn. Let X_n be the number of red marbles in the urn after n iterations of this procedure, and let Y_n=X_n/(n+r+b). Then the sequence { Y_n : n = 1, 2, 3, ... } is a martingale.

• (Likelihood-ratio testing in statistics) A population is thought to be distributed according either to a probability density f or another probability density g. A random sample is taken, the data being X₁, ... , X_n. Let Y_n be the "likelihood ratio"

(which, in applications, would be used as a test statistic). If the population is actually distributed according to the density f rather than according to g, then { Y_n : n = 1, 2, 3, ... } is a martingale with respect to { X_n : n = 1, 2, 3, ... }.

• Suppose each ameba either splits into two amebas, with probability p, or eventually dies, with probability 1 − p. Let X_n be the number of amebas surviving in the nth generation (in particular X_n = 0 if the population has become extinct by that time). Let r be the probability of eventual extinction. (Finding r as function of p is an instructive exercise. Hint: The probability that the descendants of an ameba eventually die out is equal to the probability that either of its immediate offspring dies out, given that the original ameba has split.) Then

is a martingale with respect to { X_n: n = 1, 2, 3, ... }.

Convergence of martingales

[This section should state a martingale convergence theorem and perhaps some applications, and give at least one example of a non-convergent martingale.]

Martingales and stopping times

A stopping time with respect to a sequence of random variables X₁, X₂, ... 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, ..., X_t. 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 his previous winnings (for example, he might leave only when he goes broke), but can't choose to go or stay based on the outcome of games that haven't been played yet.

Some mathematicians defined the concept of stopping time by requiring only that the occurrence or non-occurrence of the event τ = t be probabilistically independent of X_t+1, X_t+2, X_t+3, ...., but not that it be 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.

The optional stopping theorem says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value. One version of the theorem is given below:

Let X₁, X₂, ... be a martingale and τ a stopping time with respect to X₁, X₂, .... If (a) Pr[τ < ∞] = 1, (b) E[τ] < ∞, and (c) there exists a constant c such that |X_i+1−X_i| ≤ c for all i; then E[X_τ] = E[X₁].

Some applications of the theorem:

• We can use it to prove the impossibility of successful betting strategies for a gambler with a finite lifetime (which gives conditions (a) and (b)) and a house limit on bets (condition (c)). Suppose that the gambler can wager up to c dollars on a fair coin flip at times 1, 2, 3, etc., winning his wager if the coin comes up heads and losing it if the coin comes up tails. Suppose further that he can quit whenever he likes, but cannot predict the outcome of gambles that haven't happened yet. Then the gambler's fortune over time is a martingale, and the time τ at which he decides to quit (or goes broke and is forced to quit) is a stopping time. So the theorem says that E[X_τ] = E[X₁]. In other words, the gambler leaves with the same amount of money on average as when he started.

• Suppose we have a random walk that goes up or down by one with equal probability on each step. Suppose further that the walk stops if it reaches 0 or m; the time at which this first occurs is a stopping time. If we happen to know that the expected time that the walk ends is finite (say, from Markov chain theory), the optional stopping theorem tells us that the expected position when we stop is equal to the initial position a. Solving a = pm + (1−p)0 for the probability p that we reach m before 0 gives p = a/m.

• Now consider a random walk that starts at 0 and stops if we reach −m or +m, and use the Y_n=X_n²−n martingale from the examples section. If τ is the time at which we first reach ±m, then 1 = E[Y₁] = E[Y_τ] = m² - E[τ]. We immediately get E[τ] = m²+1.

Submartingales and supermartingales

A submartingale is like a martingale, except that the current value of the random variable is always less than or equal to the expected future value. Formally, this means

Similarly, in a supermartingale, the current value is always greater than or equal to the expected future value:

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 (which also follows from the fact that X²_n − n is a martingale). Similarly, a concave function of a martingale is a supermartingale.