Martingale (probability Theory) - Examples of Martingales

Examples of Martingales

• An unbiased random walk (in any number of dimensions) is an example of a martingale.

• A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair.

• Polya's urn contains a number of different coloured marbles, and each iteration a marble is randomly selected out of the urn and replaced with several more of that same colour. For any given colour, the ratio of marbles inside the urn with that colour is a martingale. For example, if currently 95% of the marbles are red then—though the next iteration is much more likely add more red marbles—this bias is exactly balanced out by the fact that adding more red marbles alters the ratio much less significantly than adding the same number of non-red marbles would.

• 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 varies roughly between plus or minus 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, ... }. To show this

• (Likelihood-ratio testing in statistics) A population is thought to be distributed according to either 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 amoeba either splits into two amoebas, with probability p, or eventually dies, with probability 1 − p. Let X_n be the number of amoebas 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 amoeba eventually die out is equal to the probability that either of its immediate offspring dies out, given that the original amoeba has split.) Then

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

• The number of individuals of any particular species in an ecosystem of fixed size is a function of (discrete) time, and may be viewed as a sequence of random variables. This sequence is a martingale under the unified neutral theory of biodiversity.

• If { N_t : t ≥ 0 } is a Poisson process with intensity λ, then the Compensated Poisson process { N_t − λt : t ≥ 0 } is a continuous-time martingale with right-continuous/left-limit sample paths.

• An example martingale series can easily be produced with computer software:

• Microsoft Excel or similar spreadsheet software. Enter 0.0 in the A1 (top left) cell, and in the cell below it (A2) enter =A1+NORMINV(RAND,0,1). Now copy that cell by dragging down to create 300 or so copies. This creates a martingale series with a mean of 0 and standard deviation of 1. With the cells still highlighted go to the chart creation tool and create a chart of these values. Now every time a recalculation happens (in Excel the F9 key does this) the chart displays another martingale series.
• R. To recreate the example above, issue plot(cumsum(rnorm(100, mean=0, sd=1)), t="l", col="darkblue", lwd=3). To display another martingale series, reissue the command.