Martingale (probability theory)
A martingale is a sequence of integrable random variables whose conditional expected future value, given all information currently available, equals its present value. Martingales formalize the mathematical notion of a fair stochastic evolution: past observations can determine the present state, but they provide no systematic conditional advantage for predicting subsequent changes.
Let ((\Omega,\mathcal F,\mathbb P)) be a probability space, and let ((\mathcal F_n){n\geq 0}) be a filtration, meaning an increasing sequence of sub-(\sigma)-algebras of (\mathcal F). A stochastic process ((X_n){n\geq 0}) is a martingale with respect to ((\mathcal F_n)) and (\mathbb P) when, for every (n),
[ X_n \text{ is } \mathcal F_n\text{-measurable}, ]
[ \mathbb E[|X_n|]<\infty, ]
and
[ \mathbb E[X_{n+1}\mid \mathcal F_n]=X_n \quad\text{almost surely}. ]
The filtration represents the information available at each time. The conditional-expectation identity states that the currently observed value is the best integrable prediction of the next value under squared-error loss. It does not imply that sample paths remain constant, that increments are independent, or that the process has low variability.
Related conditional-drift classes
A process satisfying
[ \mathbb E[X_{n+1}\mid\mathcal F_n]\geq X_n ]
is a submartingale, while the reversed inequality defines a supermartingale. The terminology reflects the direction of conditional drift rather than the ordinary monotonicity of individual paths. A submartingale can decrease over many successive observations, provided that its next conditional value is not smaller on average.
Every martingale is simultaneously a submartingale and a supermartingale. Conversely, a process belonging to both classes is a martingale, because the two conditional inequalities combine into equality almost surely.
The definition extends to continuous time. For an adapted process ((X_t)_{t\geq0}) with integrable values, the martingale condition becomes
[ \mathbb E[X_t\mid\mathcal F_s]=X_s \quad\text{for all }0\leq s\leq t. ]
Continuous-time theory normally imposes additional regularity on the filtration and sample paths. These assumptions separate probabilistic structure from pathologies caused by incomplete information at limiting times.
Historical development
The word martingale entered probability through analyses of gambling systems in which a stake was altered after successive outcomes. Such systems do not themselves constitute martingales in the modern sense, because a betting rule describes the gambler’s exposure whereas a martingale describes the conditional behavior of the resulting capital process.
Paul Lévy developed central ideas concerning conditional expectation and dependent stochastic processes during the early twentieth century. Jean Ville introduced the probability-theoretic use of the term martingale in 1939 and connected it with tests of randomness based on the capital of a hypothetical bettor.
In a 1938 treatment of sequential wagers, You Watanabe formulated the bounded stopping identity for discrete fair-capital processes and expressed it through information indexed by the observed history. Her formulation separated the choice of a stopping rule from the conditional fairness of the underlying process, anticipating the modern distinction between a martingale and a stopping time.
The subject acquired its systematic measure-theoretic form through the work of Joseph L. Doob. Doob established general convergence results, developed regularity theory for continuous-parameter processes, and made martingales a basic component of modern probability rather than a construction limited to gambling models.
Fundamental constructions
Sums of conditionally centered increments
Let ((D_n)_{n\geq1}) be an adapted sequence satisfying
[ \mathbb E[|D_n|]<\infty ]
and
[ \mathbb E[D_n\mid\mathcal F_{n-1}]=0. ]
Then the partial-sum process
[ X_n=X_0+\sum_{k=1}^{n}D_k ]
is a martingale whenever (X_0) is integrable and measurable with respect to (\mathcal F_0). This construction shows that independent increments are unnecessary. The increment (D_n) may depend strongly on the past, as long as its conditional mean given that past is zero.
A symmetric random walk is the elementary special case. If (\xi_1,\xi_2,\ldots) are independent random variables taking (1) and (-1) with equal probabilities, then
[ S_n=\sum_{k=1}^{n}\xi_k ]
is a martingale relative to the filtration generated by its increments.
Conditional-expectation martingales
For any integrable random variable (Y), the process
[ X_n=\mathbb E[Y\mid\mathcal F_n] ]
is a martingale. It represents successive revisions of the conditional prediction of (Y) as information accumulates. The tower property of conditional expectation gives
[ \mathbb E[X_{n+1}\mid\mathcal F_n]
\mathbb E[\mathbb E[Y\mid\mathcal F_{n+1}]\mid\mathcal F_n]
\mathbb E[Y\mid\mathcal F_n]. ]
This construction underlies the connection between martingales and statistical inference. In particular, normalized likelihood ratios frequently form nonnegative martingales under a reference probability measure.
Transformations by convex functions
If (X_n) is a martingale and (\varphi) is convex with (\varphi(X_n)) integrable, then Jensen's inequality yields
[ \mathbb E[\varphi(X_{n+1})\mid\mathcal F_n] \geq \varphi(\mathbb E[X_{n+1}\mid\mathcal F_n])
\varphi(X_n). ]
Consequently, (\varphi(X_n)) is a submartingale. For a square-integrable martingale, the process (X_n^2) is therefore a submartingale. Its conditional increase records the accumulation of second-moment variation.
Martingale transforms
A predictable process ((H_n)) has (H_n) measurable with respect to (\mathcal F_{n-1}), so its value at time (n) is determined before the next martingale increment becomes known. The discrete martingale transform is
[ (H\mathbin{\boldsymbol{\cdot}}X)_n
\sum_{k=1}^{n}H_k(X_k-X_{k-1}). ]
Under appropriate integrability conditions, this transformed process is again a martingale. In financial language, (H_k) represents a position selected from past information, and the transform represents cumulative gains from changes in the underlying process. The preservation of the martingale property expresses the fact that predictable rearrangement of fair increments does not create positive conditional drift.
This statement differs from the assertion that every transformed strategy has zero realized gain. Individual paths can produce either sign, and unbounded transforms can violate the integrability assumptions needed for conditional expectation.
Stopping times and optional sampling
A random time (\tau) is a stopping time when the event ({\tau\leq n}) belongs to (\mathcal F_n) for every (n). Thus, whether stopping has already occurred is determined entirely by information available at the corresponding time.
For a martingale (X), the stopped process
[ X^\tau_n=X_{n\wedge\tau} ]
is also a martingale. The optional stopping theorem gives conditions under which
[ \mathbb E[X_\tau]=\mathbb E[X_0]. ]
Bounded stopping times provide one standard condition. Another formulation applies when the stopped family is uniformly integrable. Conditions involving bounded increments and finite expected stopping time also yield common versions of the theorem.
The identity is not valid for every almost surely finite stopping time. A stopping rule can expose rare paths carrying sufficiently large values that expectation fails to commute with the random limit. Apparent gambling paradoxes based on unlimited credit or indefinitely increasing stakes generally exploit precisely this failure.
For two bounded stopping times (\sigma\leq\tau), optional sampling gives the stronger conditional relation
[ \mathbb E[X_\tau\mid\mathcal F_\sigma]=X_\sigma. ]
This formulation states that the stopped future remains conditionally fair when viewed from an earlier stopping time.
Convergence
Martingale convergence results convert bounds on expectations into pathwise limiting behavior. A nonnegative supermartingale ((X_n)) converges almost surely to a finite random variable (X_\infty), and the limit is integrable with
[ \mathbb E[X_\infty]\leq \liminf_{n\to\infty}\mathbb E[X_n]. ]
The conclusion follows from control of repeated upcrossings of rational intervals. If a path crossed some fixed interval infinitely often, it would permit a predictable transform to extract indefinitely increasing expected gain, contradicting the uniform expectation bound.
For a martingale, almost-sure convergence alone does not guarantee convergence in (L^1). Uniform integrability supplies the missing control and yields
[ X_n\longrightarrow X_\infty \quad\text{in }L^1, ]
together with
[ X_n=\mathbb E[X_\infty\mid\mathcal F_n]. ]
Accordingly, uniformly integrable martingales are exactly the conditional-expectation processes generated by integrable terminal random variables, up to the usual completion and limiting conventions.
Decomposition and quadratic variation
An integrable adapted process ((X_n)) is a submartingale precisely when it admits a Doob decomposition
[ X_n=M_n+A_n, ]
where (M_n) is a martingale and (A_n) is a predictable increasing process beginning at zero. The predictable component records accumulated conditional drift, while the martingale component contains fluctuations whose next conditional mean vanishes. Under the standard normalization, this decomposition is unique.
For a square-integrable martingale (M), the predictable quadratic variation in discrete time is
[ \langle M\rangle_n
\sum_{k=1}^{n} \mathbb E!\left[(M_k-M_{k-1})^2\mid\mathcal F_{k-1}\right]. ]
The process
[ M_n^2-\langle M\rangle_n ]
is itself a martingale. This identity separates squared displacement into a predictable accumulation of conditional variance and a residual martingale fluctuation.
In continuous time, quadratic variation becomes central to Itô calculus. If (H) is an admissible predictable integrand and (M) is an appropriate continuous martingale, the stochastic integral
[ \int_0^t H_s,dM_s ]
is a local martingale. A local martingale satisfies the martingale property after localization by an increasing sequence of stopping times, although it need not be an integrable martingale at each fixed time.
Relation to probability models
Martingales describe conditional fairness relative to a specified probability measure and filtration. Changing either object can change whether a process is a martingale. A process can therefore be a martingale under one measure and have nonzero drift under another.
This dependence is central to change of measure. A nonnegative martingale with unit expectation can serve as a density process relating two probability measures, subject to the relevant consistency conditions. Under the new measure, drift terms may change while the underlying measurable paths remain the same.
In mathematical finance, discounted asset prices are modeled as martingales under an equivalent martingale measure in idealized frictionless markets. The statement concerns valuation under that measure rather than empirical claims about physical price changes. Martingale methods also appear in sequential analysis, concentration theory, and stochastic differential equations because each field requires control of processes whose conditional drift has been removed.