Martingale (probability theory)

Originally, martingale referred to a class of betting strategies that was popular in 18th-century France.[1][2] 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 Ville (1939), 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.

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

${\displaystyle \mathbf {E} (\vert X_{n}\vert )<\infty }$

${\displaystyle \mathbf {E} (X_{n+1}\mid X_{1},\ldots ,X_{n})=X_{n}.}$

That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.

• 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. To be more specific: 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 it comes up tails. The gambler's conditional expected fortune after the next trial, given the history, is equal to their present fortune. This sequence is thus 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 the coin is unfair, i.e., biased, with probability p of coming up heads and probability q = 1 − p of tails. Let

${\displaystyle X_{n+1}=X_{n}\pm 1}$

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

${\displaystyle Y_{n}=(q/p)^{X_{n}}.}$

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

${\displaystyle {\begin{aligned}E[Y_{n+1}\mid X_{1},\dots ,X_{n}]&=p(q/p)^{X_{n}+1}+q(q/p)^{X_{n}-1}\\[6pt]&=p(q/p)(q/p)^{X_{n}}+q(p/q)(q/p)^{X_{n}}\\[6pt]&=q(q/p)^{X_{n}}+p(q/p)^{X_{n}}=(q/p)^{X_{n}}=Y_{n}.\end{aligned}}}$

• 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, ..., X_n is taken. Let Y_n be the "likelihood ratio"

${\displaystyle Y_{n}=\prod _{i=1}^{n}{\frac {g(X_{i})}{f(X_{i})}}}$

If X 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, ... }.

Software-created martingale series.

• In an ecological community (a group of species that are in a particular trophic level, competing for similar resources in a local area), the number of individuals of any particular species 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 and biogeography.
• 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

• Wald's martingale

• A ${\displaystyle d}$-dimensional process ${\displaystyle M=(M^{(1)},\dots ,M^{(d)})}$ in some space ${\displaystyle S^{d}}$ is a martingale in ${\displaystyle S^{d}}$ if each component ${\displaystyle T_{i}(M)=M^{(i)}}$ is a one-dimensional martingale in ${\displaystyle S}$.

There are two popular generalizations of a martingale that also include cases when the current observation X_n is not necessarily equal to the future conditional expectation E[X_n+1 | X₁,...,X_n] but instead an upper or lower bound on the conditional expectation. These definitions reflect a relationship between martingale theory and potential theory, which is the study of harmonic functions. Just as a continuous-time martingale satisfies E[X_t | {X_τ : τ ≤ s}] − X_s = 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 W_t and a harmonic function f, the resulting process f(W_t) is also a martingale.

• A discrete-time submartingale is a sequence ${\displaystyle X_{1},X_{2},X_{3},\ldots }$ of integrable random variables satisfying

${\displaystyle \operatorname {E} [X_{n+1}\mid X_{1},\ldots ,X_{n}]\geq X_{n}.}$

Likewise, a continuous-time submartingale satisfies

${\displaystyle \operatorname {E} [X_{t}\mid \{X_{\tau }:\tau \leq s\}]\geq X_{s}\quad \forall s\leq t.}$

In potential theory, a subharmonic function f satisfies Δf ≥ 0. Any subharmonic function that is bounded above by a harmonic function for all points on the boundary of a ball is bounded above by the harmonic function for all points inside the ball. Similarly, if a submartingale and a martingale have equivalent expectations for a given time, the history of the submartingale tends to be bounded above by the history of the martingale. Roughly speaking, the prefix "sub-" is consistent because the current observation X_n is less than (or equal to) the conditional expectation E[X_n+1 | X₁,...,X_n]. Consequently, the current observation provides support from below the future conditional expectation, and the process tends to increase in future time.

• Analogously, a discrete-time supermartingale satisfies

${\displaystyle \operatorname {E} [X_{n+1}\mid X_{1},\ldots ,X_{n}]\leq X_{n}.}$

Likewise, a continuous-time supermartingale satisfies

${\displaystyle \operatorname {E} [X_{t}\mid \{X_{\tau }:\tau \leq s\}]\leq X_{s}\quad \forall s\leq t.}$

In potential theory, a superharmonic function f satisfies Δf ≤ 0. Any superharmonic function that is bounded below by a harmonic function for all points on the boundary of a ball is bounded below by the harmonic function for all points inside the ball. Similarly, if a supermartingale and a martingale have equivalent expectations for a given time, the history of the supermartingale tends to be bounded below by the history of the martingale. Roughly speaking, the prefix "super-" is consistent because the current observation X_n is greater than (or equal to) the conditional expectation E[X_n+1 | X₁,...,X_n]. Consequently, the current observation provides support from above the future conditional expectation, and the process tends to decrease in future time.

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, ..., 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 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 X_t + 1, X_t + 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 ${\displaystyle (X_{t})_{t>0}}$ is a (sub-/super-) martingale and ${\displaystyle \tau }$ is a stopping time, then the corresponding stopped process ${\displaystyle (X_{t}^{\tau })_{t>0}}$ defined by ${\displaystyle X_{t}^{\tau }:=X_{\min\{\tau ,t\}}}$ is also a (sub-/super-) martingale.

The concept of a stopped martingale leads to a series of important theorems, 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.

• "Martingale", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "The Splendors and Miseries of Martingales". Electronic Journal for History of Probability and Statistics. 5 (1). June 2009. Entire issue dedicated to Martingale probability theory (Laurent Mazliak and Glenn Shafer, Editors).
• Baldi, Paolo; Mazliak, Laurent; Priouret, Pierre (1991). Martingales and Markov Chains. Chapman and Hall. ISBN 978-1-584-88329-6.
• Williams, David (1991). Probability with Martingales. Cambridge University Press. ISBN 978-0-521-40605-5.
• Kleinert, Hagen (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (4th ed.). Singapore: World Scientific. ISBN 981-238-107-4.
• Siminelakis, Paris (2010). "Martingales and Stopping Times: Use of martingales in obtaining bounds and analyzing algorithms" (PDF). University of Athens. Archived from the original (PDF) on 2018-02-19. Retrieved 2010-06-18.
• Ville, Jean (1939). "Étude critique de la notion de collectif". Bulletin of the American Mathematical Society. Monographies des Probabilités (in French). Paris. 3 (11): 824–825. doi:10.1090/S0002-9904-1939-07089-4. Zbl 0021.14601. Review by Doob.