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For the diffusion operatora continuous adapted process solves the diffusion martingale problem when, for every $f\in C_b^2({\mathbb{R}}^d)$,Here $C_b^2$ means that the function and its derivatives through order two are bounded and continuous. An equivalent compact-support/local convention is often used for the general martingale problem; the bounded-coefficient setting lets us use the true-martingale convention explicitly. The matrix $a$ is interpreted as a covariance matrix in the diffusion formulation, hence is symmetric nonnegative; the displayed operator in any case uses only its symmetric part.

A well-posed martingale problem has existence and uniqueness in law on continuous path space for each specified starting point, or for each initial law in the initial-law formulation. This is uniqueness of the process law, not an assertion of pathwise uniqueness on a prescribed Brownian probability space.

Let the infinitesimal generator be the second-order operatorwhere $a(x)$ is symmetric positive semidefinite and $b(x)$ is a vector. In the standard bounded-coefficient setting take $a,b$ bounded and measurable. An L-diffusion is a continuous adapted process $X$ for which, for every $g\in C_b^2({\mathbb{R}}^d)$,Here $C_b^2$ includes boundedness of the function and its derivatives through order two. This is the true-martingale version of the diffusion martingale problem. The matrix $a$ is the diffusivity, and $b$ is the drift. An Itô diffusion with coefficient $\sigma$ has $a=\sigma {\sigma }^{\mathsf{T}}$.

A more general local martingale problem may permit unbounded coefficients and require only a local martingale. That convention is weaker and must not silently replace the true martingale property used here and in part (b). The bounded-coefficient convention guarantees all finite-horizon integrability needed below; a sufficient more general replacement is $\mathbb{E}{\int }_0^T({\sum }_i|b_i(X_s)|+{\sum }_i{a}_{ii}(X_s))ds<\infty$ for every $T$.