Inprobability theory, amartingale is asequence ofrandom variables (i.e., astochastic process) for which, at a particular time, theconditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.

Originally,martingale referred to a class ofbetting strategies that was popular in 18th-centuryFrance.^[1][2] The simplest of these strategies was designed for a game in which thegambler 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 asure thing. However, theexponential 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 byPaul Lévy in 1934, though he did not name it. The term "martingale" was introduced later byVille (1939), who also extended the definition to continuous martingales. Much of the original development of the theory was done byJoseph 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 adiscrete-timemartingale is a discrete-timestochastic process (i.e., asequence ofrandom variables)X₁, X₂, X₃, ... that satisfies for any timen,

${\displaystyle \mathbf{E}(X_{n+1}\mid X_1,\dots ,X_n)=X_n.}$

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

More generally, a sequenceY₁, Y₂, Y₃ ... is said to be amartingale with respect to another sequenceX₁, X₂, X₃ ... if for alln

${\displaystyle \mathbf{E}(Y_{n+1}\mid X_1,\dots ,X_n)=Y_n.}$

Similarly, acontinuous-time martingale with respect to thestochastic processX_t is astochastic processY_t such that for allt

${\displaystyle \mathbf{E}(Y_t\mid \{X_{\tau },\tau \le s\})=Y_s\mathrm{\forall }s\le t.}$

This expresses the property that the conditional expectation of an observation at timet, given all the observations up to time${\displaystyle s}$, is equal to the observation at times (of course, provided thats ≤ t). The second property implies that${\displaystyle Y_n}$ is measurable with respect to${\displaystyle X_1\dots X_n}$.

In full generality, astochastic process${\displaystyle Y:T\times \mathrm{\Omega }\to S}$ taking values in aBanach space${\displaystyle S}$ with norm${\displaystyle \Vert \cdot {\Vert }_S}$ is amartingale with respect to a filtration${\displaystyle {\mathrm{\Sigma }}_{\ast }}$andprobability measure${\displaystyle \mathbb{P}}$ if

• Σ_∗ is afiltration of the underlyingprobability space (Ω, Σ, ${\displaystyle \mathbb{P}}$);
• Y isadapted to the filtration Σ_∗, i.e., for eacht in theindex setT, the random variableY_t is a Σ_t-measurable function;
• for eacht,Y_t lies in theL^p spaceL¹(Ω, Σ_t, ${\displaystyle \mathbb{P}}$; S), i.e.

• for alls andt withs < t and allF ∈ Σ_s,

${\displaystyle {\mathbf{E}}_{\mathbb{P}}\left([Y_t-Y_s]{\chi }_F\right)=0,}$

whereχ_F denotes theindicator function of the eventF. In Grimmett and Stirzaker'sProbability and Random Processes, this last condition is denoted as

${\displaystyle Y_s={\mathbf{E}}_{\mathbb{P}}(Y_t\mid {\mathrm{\Sigma }}_s),}$

which is a general form ofconditional expectation.^[3]

It is important to note that the property of being a martingale involves both the filtrationand the probability measure (with respect to which the expectations are taken). It is possible thatY could be a martingale with respect to one measure but not another one; theGirsanov theorem offers a way to find a measure with respect to which anItō process is a martingale.

In the Banach space setting the conditional expectation is also denoted in operator notation as${\displaystyle {\mathbf{E}}^{{\mathrm{\Sigma }}_s}{Y}_t}$.^[4]

• An unbiasedrandom 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 (capital) is a martingale if all the betting games which the gambler plays are fair. The gambler is playing a game ofcoin flipping. SupposeX_n is the gambler's fortune aftern tosses of afair 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.
• LetY_n =X_n² −n whereX_n is the gambler's fortune from the prior 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 thesquare root of the number of games of coin flipping played.
• de Moivre's martingale: Suppose thecoin toss outcomes are unfair, i.e., biased, with probabilityp of coming up heads and probabilityq = 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

• Pólya's urn contains a number of different-coloured marbles; at eachiteration 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 instatistics: A random variableX is thought to be distributed according either to probability densityf or to a different probability densityg. Arandom sampleX₁, ...,X_n is taken. LetY_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 densityf rather than according tog, then {Y_n :n=1, 2, 3,...} is a martingale with respect to {X_n :n=1, 2, 3, ...}

• In anecological community, i.e. 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 theunified neutral theory of biodiversity and biogeography.
• If {N_t :t ≥ 0 } is aPoisson process with intensityλ, then the compensated Poisson process { N_t − λt :t ≥ 0 } is a continuous-time martingale withright-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 generalizations of a martingale that also include cases when the current observationX_n is not necessarily equal to the future conditional expectationE[X_n+1 | X₁,...,X_n] but instead an upper or lower bound on the conditional expectation. These generalizations reflect the relationship between martingale theory andpotential theory, that is, the study ofharmonic functions. Just as a continuous-time martingale satisfies E[X_t | {X_τ : τ ≤ s}] − X_s = 0 ∀s ≤ t, a harmonic functionf satisfies thepartial differential equation Δf = 0 where Δ is theLaplacian operator. Given aBrownian motion processW_t and a harmonic functionf, the resulting processf(W_t) is also a martingale.

• A discrete-timesubmartingale is a sequence${\displaystyle X_1,X_2,X_3,\dots }$ ofintegrable random variables satisfying

${\displaystyle \mathrm{E}[X_{n+1}\mid X_1,\dots ,X_n]\ge X_n.}$

Likewise, a continuous-time submartingale satisfies

${\displaystyle \mathrm{E}[X_t\mid \{X_{\tau }:\tau \le s\}]\ge X_s\mathrm{\forall }s\le t.}$

In potential theory, asubharmonic functionf 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, theprefix "sub-" is consistent because the current observationX_n isless than (or equal to) the conditional expectationE[X_n+1 | X₁,...,X_n]. Consequently, the current observation provides supportfrom below the future conditional expectation, and the process tends to increase in future time.

• Analogously, a discrete-timesupermartingale satisfies

${\displaystyle \mathrm{E}[X_{n+1}\mid X_1,\dots ,X_n]\le X_n.}$

Likewise, a continuous-time supermartingale satisfies

${\displaystyle \mathrm{E}[X_t\mid \{X_{\tau }:\tau \le s\}]\le X_s\mathrm{\forall }s\le t.}$

In potential theory, asuperharmonic functionf 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 observationX_n isgreater than (or equal to) the conditional expectationE[X_n+1 | X₁,...,X_n]. Consequently, the current observation provides supportfrom above the future conditional expectation, and the process tends to decrease in future time.

• Every martingale is also a submartingale and a supermartingale. Conversely, any stochastic process that isboth 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 probabilityp.
  • Ifp is equal to 1/2, the gambler on average neither wins nor loses money, and the gambler's fortune over time is a martingale.
  • Ifp is less than 1/2, the gambler loses money on average, and the gambler's fortune over time is a supermartingale.
  • Ifp is greater than 1/2, the gambler wins money on average, and the gambler's fortune over time is a submartingale.
• Aconvex function of a martingale is a submartingale, byJensen'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 thatX_n² − n is a martingale). Similarly, aconcave function of a martingale is a supermartingale.

Astopping time with respect to a sequence of random variablesX₁, X₂, X₃, ... is a random variable τ with the property that for eacht, the occurrence or non-occurrence of the eventτ =t depends only on the values ofX₁, X₂, X₃, ..., X_t. The intuition behind the definition is that at any particular timet, 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 ofstopping time is defined by requiring only that the occurrence or non-occurrence of the eventτ = t isprobabilistically independent ofX_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, theoptional 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.
• Richard, Mark; Vecer, Jan (2021)."Efficiency Testing of Prediction Markets: Martingale Approach, Likelihood Ratio and Bayes Factor Analysis".Risks.9 (2): 31.doi:10.3390/risks9020031.hdl:10419/258120.
• Siminelakis, Paris (2010)."Martingales and Stopping Times: Use of martingales in obtaining bounds and analyzing algorithms"(PDF). University of Athens. Archived fromthe original(PDF) on 2018-02-19. Retrieved2010-06-18.
• Ville, Jean (1939)."Étude critique de la notion de collectif".Bulletin of the American Mathematical Society. Monographies des Probabilités (in French).3 (11). Paris:824–825.doi:10.1090/S0002-9904-1939-07089-4.Zbl 0021.14601.Review by Doob.

• Stochastic processes
• Martingale theory
• Game theory
• Paul Lévy (mathematician)

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