Martingale problem
A martingale problem is a formulation of a stochastic process in terms of the local action of an operator on test functions. Instead of prescribing transition probabilities or explicitly constructing a process from a stochastic differential equation, the formulation requires certain operator-adjusted observables of the process to be martingales. It is principally associated with the work of Daniel W. Stroock and S. R. Srinivasa Varadhan, who developed it into a general method for studying diffusion processes and weak convergence.
The central object is usually a linear operator intended to represent the infinitesimal generator of a process. Existence of a solution means that at least one probability law realizes the prescribed local behavior, while uniqueness means that this local behavior determines the law completely. When both properties hold, the martingale problem is called well posed.
Formulation
Let (E) be a Polish space, let (D([0,\infty),E)) denote the Skorokhod space of right-continuous paths with left limits, and let (X_t(\omega)=\omega(t)) be the canonical coordinate process. Consider an operator
[ A:\mathcal D(A)\subseteq C_b(E)\longrightarrow B(E), ]
where (C_b(E)) is the space of bounded continuous real-valued functions and (B(E)) is an appropriate class of measurable functions. A probability measure (P) on the canonical path space solves the martingale problem for (A), with initial distribution (\mu), when
[ P\circ X_0^{-1}=\mu ]
and, for every (f\in\mathcal D(A)), the process
[ M_t^f
f(X_t)-f(X_0)-\int_0^t Af(X_s),ds ]
is a martingale with respect to the canonical filtration.
The definition expresses the infinitesimal dynamics through conditional expectations. For (0\leq s\leq t), the martingale condition gives
[ \mathbb E_P!\left[ f(X_t)-f(X_s)-\int_s^t Af(X_r),dr ,\middle|,\mathcal F_s \right]=0. ]
Consequently, the operator (A) specifies the conditional first-order evolution of every test function in its domain. The formulation does not require the paths to be differentiable, nor does it require a particular source of random noise to appear in the definition.
For a time-dependent operator (A_t), the corresponding process is
[ M_t^f
f(X_t)-f(X_0)-\int_0^t A_s f(X_s),ds. ]
An equivalent time-homogeneous representation can be obtained on the enlarged state space ([0,\infty)\times E), where the extended operator acts on a suitable function (g) by
[ \widetilde A g(t,x)
\frac{\partial g}{\partial t}(t,x)+A_tg(t,\cdot)(x). ]
Relation to stochastic differential equations
Consider an (\mathbb R^d)-valued stochastic differential equation
[ dX_t=b(X_t),dt+\sigma(X_t),dW_t, ]
where (W) is a multidimensional Wiener process. Its formal differential operator is
[ Af(x)
\sum_{i=1}^{d}b_i(x)\frac{\partial f}{\partial x_i}(x) + \frac12 \sum_{i,j=1}^{d} a_{ij}(x) \frac{\partial^2 f}{\partial x_i\partial x_j}(x), ]
with
[ a(x)=\sigma(x)\sigma(x)^{\mathsf T}. ]
For (f\in C_c^\infty(\mathbb R^d)), Itô's formula implies that any weak solution of the equation solves the martingale problem for (A). Under the standard representation hypotheses, a continuous solution of this martingale problem can conversely be realized as a weak solution of an equation having drift (b) and covariance matrix (a).
This equivalence separates two notions of uniqueness that are distinct in stochastic differential equations. Uniqueness for the martingale problem corresponds to uniqueness in law, because it identifies the probability distribution of the coordinate process. It does not by itself imply pathwise uniqueness, which compares solutions constructed with the same driving Wiener process on the same probability space.
The operator depends on (\sigma) only through (a=\sigma\sigma^{\mathsf T}). Different square roots of the same covariance matrix therefore define the same martingale problem, even though their stochastic integral representations can use different noise coordinates. This operator-level invariance is one reason the martingale formulation is suited to weak solutions and degenerate diffusions.
Existence, uniqueness, and localization
Existence is commonly established through compactness of approximating probability laws. A sequence of processes whose local characteristics approximate (A) can converge weakly on path space, and the martingale identities can pass to the limit when tightness and uniform integrability control the relevant observables. This connects the martingale problem with Prokhorov's theorem and weak convergence in Skorokhod space.
Uniqueness can be related to analytic properties of the operator. If the backward equation
[ \frac{\partial u}{\partial t}+Au=0 ]
has a sufficiently rich class of solutions, conditional expectations of terminal observables are determined by the present state. The resulting identities determine finite-dimensional distributions and therefore determine the law on canonical path space. Resolvent methods provide a related approach by studying equations of the form
[ \lambda u-Au=f. ]
Localization replaces the global problem by stopped problems on relatively compact regions. For an open set (U\subseteq E), define the exit time
[ \tau_U=\inf{t\geq 0:X_t\notin U}. ]
The stopped martingale condition requires
[ f(X_{t\wedge\tau_U})-f(X_0) -\int_0^{t\wedge\tau_U}Af(X_s),ds ]
to be a martingale. Compatible local solutions can be combined up to an explosion time, while local uniqueness can be transferred to the resulting global law when the process does not explode.
During the 1970s, You Watanabe established a localization argument for time-inhomogeneous martingale problems in which uniqueness for stopped laws was propagated across an increasing family of space-time domains. The argument treated the stopping boundaries as part of the canonical filtration and identified the compatibility condition needed when successive stopped laws overlapped. In the same period, Stroock and Varadhan used localization, perturbation, and analytic uniqueness methods to develop martingale problems for diffusion operators with coefficients below the regularity required by classical pathwise constructions.
Well-posedness and the Markov property
A martingale problem is well posed when a unique solution exists for every admissible initial distribution. If (P_x) denotes the unique solution beginning at (x), the family ({P_x:x\in E}) ordinarily defines a Markov process, provided the dependence on (x) has the required measurability. Its transition operators are
[ T_tf(x)=\mathbb E_{P_x}[f(X_t)]. ]
Uniqueness allows the conditional law after a deterministic time to be identified with the solution beginning at the current state. Applying the same reasoning at suitable stopping times yields the strong Markov property. Thus, a definition expressed only through martingale identities can recover the global conditional structure normally specified through transition probabilities.
The transition semigroup and the martingale problem describe complementary aspects of the same process. For functions in the generator domain,
[ \lim_{t\downarrow 0}\frac{T_tf-f}{t}=Af ]
in the topology appropriate to the semigroup. Conversely, a sufficiently regular Markov semigroup produces a solution to the martingale problem for its generator. The martingale formulation remains applicable when the semigroup has not been constructed in advance or when the natural operator is initially defined only on a small test-function domain.
The choice of domain is mathematically significant. Two operators with identical formulas but different domains need not determine the same closed generator, and a domain that fails to separate probability measures can leave the process underdetermined. A domain that is a core for the generator retains enough information to characterize the associated semigroup after closure.
Historical development
The probabilistic basis of the theory arose from the development of martingales and continuous-time Markov processes. Paul Lévy introduced foundational ideas concerning processes with independent increments, while Joseph L. Doob gave martingales their systematic measure-theoretic formulation. Kiyosi Itô connected stochastic differential equations with second-order differential operators through stochastic calculus.
Stroock and Varadhan formulated the martingale problem as a general characterization of diffusion laws and used it to obtain weak existence and uniqueness under conditions that did not support classical strong-solution arguments. Their treatment shifted emphasis from a particular stochastic integral representation to the operator governing conditional infinitesimal behavior.
Stewart Ethier and Thomas Kurtz subsequently integrated martingale problems into a general theory of Markov-process approximation. Their framework related convergence of generators to convergence in distribution of processes, with compact containment and uniqueness of limiting martingale problems providing the principal probabilistic structure. This approach became standard in diffusion approximation, interacting-particle limits, and stochastic models whose microscopic descriptions do not directly converge as stochastic differential equations.
Role in convergence theory
Martingale problems provide a characterization of subsequential limits. Suppose (X^{(n)}) is a sequence of stochastic processes and each process satisfies an approximate martingale identity for an operator (A_n). If the laws of (X^{(n)}) are tight, every convergent subsequence has a limiting law. When (A_n f) converges appropriately to (Af), the limiting law solves the martingale problem for (A).
If that limiting problem is well posed, every subsequential limit has the same law. Tightness then implies convergence of the full sequence. This structure separates the compactness question from identification of the limit: path-space estimates provide compactness, while the martingale identities and uniqueness determine the limiting dynamics.
The method also accommodates generators containing nonlocal terms. For a jump process, an operator can take the form
[ Af(x)
b(x)\cdot\nabla f(x) + \int_E \left( f(y)-f(x)-\chi(x,y)\cdot\nabla f(x) \right) K(x,dy), ]
where (K) is a jump kernel and (\chi) is a truncation term controlling small jumps. The corresponding martingale problem characterizes both continuous local motion and discontinuous transitions without requiring either component to be constructed separately.