the martingale problem
/ STROHK vah-rah-DAHN /
The martingale problem is Stroock and Varadhan's way of DEFINING a Markov process directly by its generator, without first constructing a transition function or solving an SDE. It is the most flexible characterization: it asks for a probability measure on path space under which a specific family of processes built from A are martingales, and it shifts the hard analytic question 'does the process exist and is it unique?' into 'does this martingale problem have a solution, and is it well-posed?'.
Setup: fix a candidate generator A (say a second-order operator with coefficients b, a) acting on a core such as C_c^infinity. A probability measure P on the path space (typically càdlàg paths) is said to SOLVE the martingale problem for A started at x if, under P, the coordinate process X has X_0 = x almost surely and, for every f in the core, M_t^f = f(X_t) - f(X_0) - integral_0^t (A f)(X_s) ds is a P-martingale with respect to the canonical filtration. EXISTENCE of a solution corresponds to the existence of the process; UNIQUENESS of the solution (well-posedness) corresponds to uniqueness in law of the process, and Stroock-Varadhan proved that well-posedness implies the solution is strong Markov and equals the diffusion with generator A. The deep theorem is that for elliptic A = (1/2) sum a_(ij) d_i d_j + sum b_i d_i with continuous, bounded coefficients and a uniformly positive-definite, the martingale problem is well-posed — a landmark existence-uniqueness result for diffusions under merely continuous (not Lipschitz) coefficients.
Why it matters: the martingale-problem formulation is robust where SDE methods struggle — degenerate or discontinuous coefficients, jumps, processes on manifolds or with boundary conditions — and it dovetails with weak convergence: to construct a process as a scaling limit, show tightness of the approximations then that any limit point solves the martingale problem, and uniqueness pins down the limit. Two honest points: well-posedness is the crux and can fail (degenerate ellipticity, bad coefficients give non-uniqueness, exactly mirroring non-uniqueness of weak SDE solutions); and 'solving the martingale problem' yields the LAW (a weak solution / uniqueness in law), not a pathwise strong solution — the Yamada-Watanabe circle of ideas relates the two.
To build a diffusion with generator A f = (1/2) a(x) f'' + b(x) f' when a, b are merely continuous (no Lipschitz bound), the SDE dX = b dt + sqrt(a) dB may lack a strong solution, yet the martingale problem for A is well-posed (for a > 0), so the diffusion exists uniquely in law. This is the Stroock-Varadhan existence theorem.
The martingale problem builds diffusions under continuous coefficients where pathwise SDE theory needs Lipschitz.
A solution gives the law (weak solution / uniqueness in law), not a strong pathwise solution; and well-posedness can fail for degenerate or sufficiently irregular A.