Martingale (probability theory)

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

Stopped Brownian motionis an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy.

History

Originally,martingalereferred to a class ofbetting strategiesthat was popular in 18th-centuryFrance.^[1]^[2]The simplest of these strategies was designed for a game in which thegamblerwins 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 growthof 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 Lévyin 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 Doobamong others. Part of the motivation for that work was to show the impossibility of successful betting strategies in games of chance.

Definitions

A basic definition of adiscrete-timemartingaleis a discrete-timestochastic process(i.e., asequenceofrandom variables)X₁,X₂,X₃,... that satisfies for any timen,

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

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

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

Martingale sequences with respect to another sequence

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

\mathbf {E} (\vert Y_{n}\vert )<\infty

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

Similarly, acontinuous-timemartingale with respect tothestochastic processX_tis astochastic processY_tsuch that for allt

\mathbf {E} (\vert Y_{t}\vert )<\infty

\mathbf {E} (Y_{t}\mid \{X_{\tau },\tau \leq s\})=Y_{s}\quad \forall s\leq t.

This expresses the property that the conditional expectation of an observation at timet,given all the observations up to time $s$ ,is equal to the observation at times(of course, provided thats≤t). The second property implies that $Y_{n}$ is measurable with respect to $X_{1}\dots X_{n}$ .

General definition

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

Σ_∗is afiltrationof the underlyingprobability space(Ω, Σ, $\mathbb {P}$ );
Yisadaptedto the filtration Σ_∗,i.e., for eachtin theindex setT,the random variableY_tis a Σ_t-measurable function;
for eacht,Y_tlies in theL^pspaceL¹(Ω, Σ_t, $\mathbb {P}$ ;S), i.e.

\mathbf {E} _{\mathbb {P} }(\lVert Y_{t}\rVert _{S})<+\infty;

for allsandtwiths<tand allF∈ Σ_s,

\mathbf {E} _{\mathbb {P} }\left([Y_{t}-Y_{s}]\chi _{F}\right)=0,

whereχ_Fdenotes theindicator functionof the eventF.In Grimmett and Stirzaker'sProbability and Random Processes,this last condition is denoted as

Y_{s}=\mathbf {E} _{\mathbb {P} }(Y_{t}\mid \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 filtrationandthe probability measure (with respect to which the expectations are taken). It is possible thatYcould be a martingale with respect to one measure but not another one; theGirsanov theoremoffers a way to find a measure with respect to which anItō processis a martingale.

In the Banach space setting the conditional expectation is also denoted in operator notation as $\mathbf {E} ^{\Sigma _{s}}Y_{t}$ .^[4]

Examples of martingales

An unbiasedrandom 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. The gambler is playing a game ofcoin flipping.SupposeX_nis the gambler's fortune afterntosses 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−nwhereX_nis 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 rootof the number of games of coin flipping played.
de Moivre's martingale: Suppose thecoin toss outcomes are unfair,i.e., biased, with probabilitypof coming up heads and probabilityq= 1 −pof tails. Let

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

{\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 urncontains a number of different-coloured marbles; at eachiterationa 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 testinginstatistics:A random variableXis thought to be distributed according either to probability densityfor to a different probability densityg.Arandom sampleX₁,...,X_nis taken. LetY_nbe the "likelihood ratio"

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

If X is actually distributed according to the densityfrather 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 processwith intensityλ,then the compensated Poisson process {N_t−λt:t≥ 0 } is a continuous-time martingale withright-continuous/left-limitsample paths.
Wald's martingale
A $d$ -dimensional process $M=(M^{(1)},\dots,M^{(d)})$ in some space $S^{d}$ is a martingale in $S^{d}$ if each component $T_{i}(M)=M^{(i)}$ is a one-dimensional martingale in $S$ .

Submartingales, supermartingales, and relationship to harmonic functions

There are two generalizations of a martingale that also include cases when the current observationX_nis 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 functionfsatisfies thepartial differential equationΔf= 0 where Δ is theLaplacian operator.Given aBrownian motionprocessW_tand a harmonic functionf,the resulting processf(W_t) is also a martingale.

A discrete-timesubmartingaleis a sequence $X_{1},X_{2},X_{3},\ldots$ ofintegrablerandom variables satisfying

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

Likewise, a continuous-time submartingale satisfies

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

In potential theory, asubharmonic functionfsatisfies 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_nisless than(or equal to) the conditional expectationE[X_n₊₁|X₁,...,X_n]. Consequently, the current observation provides supportfrom belowthe future conditional expectation, and the process tends to increase in future time.

Analogously, a discrete-timesupermartingalesatisfies

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

Likewise, a continuous-time supermartingale satisfies

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

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

Examples of submartingales and supermartingales

Every martingale is also a submartingale and a supermartingale. Conversely, any stochastic process that isbotha 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.
- Ifpis equal to 1/2, the gambler on average neither wins nor loses money, and the gambler's fortune over time is a martingale.
- Ifpis less than 1/2, the gambler loses money on average, and the gambler's fortune over time is a supermartingale.
- Ifpis greater than 1/2, the gambler wins money on average, and the gambler's fortune over time is a submartingale.
Aconvex functionof 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−nis a martingale). Similarly, aconcave functionof a martingale is a supermartingale.

Martingales and stopping times

Astopping timewith 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=tdepends 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 timeis defined by requiring only that the occurrence or non-occurrence of the eventτ=tisprobabilistically independentofX_{t+ 1},X_{t+ 2},... but not that it is completely determined by the history of the process up to timet.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, including, for example, theoptional stopping theoremwhich states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.

Notes

^Balsara, N. J. (1992).Money Management Strategies for Futures Traders.Wiley Finance. p.122.ISBN 978-0-471-52215-7.martingale.
^Mansuy, Roger (June 2009)."The origins of the Word" Martingale ""(PDF).Electronic Journal for History of Probability and Statistics.5(1).Archived(PDF)from the original on 2012-01-31.Retrieved2011-10-22.
^Grimmett, G.; Stirzaker, D. (2001).Probability and Random Processes(3rd ed.). Oxford University Press.ISBN 978-0-19-857223-7.
^Bogachev, Vladimir (1998).Gaussian Measures.American Mathematical Society. pp. 372–373.ISBN 978-1470418694.

References

"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.

[1] Balsara, N. J. (1992).Money Management Strategies for Futures Traders.Wiley Finance. p.122.ISBN 978-0-471-52215-7.martingale.

[2] Mansuy, Roger (June 2009)."The origins of the Word" Martingale ""(PDF).Electronic Journal for History of Probability and Statistics.5(1).Archived(PDF)from the original on 2012-01-31.Retrieved2011-10-22.

[3] Grimmett, G.; Stirzaker, D. (2001).Probability and Random Processes(3rd ed.). Oxford University Press.ISBN 978-0-19-857223-7.

[4] Bogachev, Vladimir (1998).Gaussian Measures.American Mathematical Society. pp. 372–373.ISBN 978-1470418694.

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