Martingale (probability theory)

A stopped Brownian motion as an example for a martingale

In probability theory, a martingale is a stochastic process (i.e., a sequence of random variables) such that the conditional expected value of an observation at some time t, given all the observations up to some earlier time s, is equal to the observation at that earlier time s. A martingale is a model of a fair game. Precise definitions are given below.

Contents

History

Originally, martingale referred to a class of betting strategies that was popular in 18th century France.[1] 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. As the gambler's wealth and available time jointly approach infinity, his 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.

The concept of martingale in probability theory was introduced by Paul Pierre Lévy, and 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.

Definitions

A discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies for all n

\mathbf{E} ( \vert X_n \vert )< \infty
\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n,

i.e., the conditional expected value of the next observation, given all the past observations, is equal to the last observation.

Somewhat 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

\mathbf{E} ( \vert Y_n \vert )< \infty
\mathbf{E} (Y_{n+1}\mid X_1,\ldots,X_n)=Y_n.

Similarly, a continuous-time martingale with respect to the stochastic process Xt is a stochastic process Yt such that for all t

\mathbf{E} ( \vert Y_t \vert )<\infty
\mathbf{E} ( Y_{t} \mid \{ X_{\tau}, \tau \leq s \} ) = Y_s, \ \forall\ s \leq t.

This expresses the property that the conditional expectation of an observation at time t, given all the observations up to time s , is equal to the observation at time s (of course, provided that s ≤ t).

In full generality, a stochastic process Y : T × Ω → S is a martingale with respect to a filtration Σ∗ and probability measure P if

\mathbf{E}_{\mathbf{P}} ( | Y_{t} | ) < + \infty;
\mathbf{E}_{\mathbf{P}} \left([Y_t-Y_s]\chi_F\right)=0,
where χF denotes the indicator function of the event F. In Grimmett and Stirzaker's Probability and Random Processes, this last condition is denoted as
Y_s = \mathbf{E}_{\mathbf{P}} ( Y_t | \Sigma_s ),
which is a general form of conditional expectation.[2]

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). 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.

Examples of martingales

X_{n+1}=X_n\pm 1
with "+" in case of "heads" and "−" in case of "tails". Let
Y_n=(q/p)^{X_n}.
Then { Yn : n = 1, 2, 3, ... } is a martingale with respect to { Xn : n = 1, 2, 3, ... }. To show this
\begin{align} E[Y_{n+1} \mid X_1,\dots,X_n] & = p (q/p)^{X_n+1} + q (q/p)^{X_n-1} \\ & = p (q/p) (q/p)^{X_n} + q (p/q) (q/p)^{X_n} \\ & = q (q/p)^{X_n} + p (q/p)^{X_n} = (q/p)^{X_n}=Y_n. \end{align}
Y_n=\prod_{i=1}^n\frac{g(X_i)}{f(X_i)}
(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 { Yn : n = 1, 2, 3, ... } is a martingale with respect to { Xn : n = 1, 2, 3, ... }.
\{\,r^{X_n}:n=1,2,3,\dots\,\}
is a martingale with respect to { Xn: n = 1, 2, 3, ... }.
Software-created martingale series.
  • 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 will create 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 will display 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.

Submartingales and supermartingales

A (discrete-time) submartingale is a sequence X_1,X_2,X_3,... of integrable random variables satisfying

{}E[X_{n+1}|X_1,\ldots,X_n] \ge X_n.

Analogously a (discrete-time) supermartingale satisfies

{}E[X_{n+1}|X_1,\ldots,X_n] \le X_n.

The more general definitions of both discrete-time and continuous-time martingales given earlier can be converted into the corresponding definitions of sub/supermartingales in the same way by replacing the equality for the conditional expectation by an inequality.

Here is a mnemonic for remembering which is which: "Life is a supermartingale; as time advances, expectation decreases."

Examples of submartingales and supermartingales

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 his 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.

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 Xt + 1, Xt + 2, ... 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.

One of the basic properties of martingales is that if (X_t)_{t>0} is a (sub-/super-) martingale and \tau is a stopping time, then the corresponding stopped process (X_t^\tau)_{t>0} defined by X_t^\tau:=X_{\min\{\tau,t\}} is also a (sub-/super-) martingale.

The concept of a stopped martingale leads to a series of important theorems. Eg. the optional stopping theorem (or optional sampling theorem), that says, under certain conditions, that the expected value of a martingale at a stopping time is equal to its initial value. We can use it, for example, to prove the impossibility of successful betting strategies for a gambler with a finite lifetime and a house limit on bets.

See also

Notes

  1. ↑ N. J. Balsara, Money Management Strategies for Futures Traders, Wiley Finance, 1992, ISBN 0-47-152215-5 pp. 122
  2. ↑ G. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001, ISBN 0-19-857223-9

References