Diffusion martingale problem ($L=\frac{1}{2}{\sum }_{i,j}{a}_{ij}{\partial }_{ij}+{\sum }_i{b}_i{\partial }_i$)

For bounded measurable $a,b$, with $a$ symmetric positive semidefinite, an L-diffusion is a continuous adapted process $X$ such that $g(X_t)-g(X_0)-{\int }_0^{t}Lg(X_s)ds$ is a true martingale for every $g\in C_b^2$. The general local martingale problem may instead use compactly supported test functions and require only a local martingale. Specify the test class and the true or local convention when coefficients are unbounded.

For an L-diffusion with bounded coefficients and $f\in C_b^{1,2}$, $M^f$ is a continuous martingale. Freeze the time argument along a deterministic partition, apply the spatial martingale problem on each interval, and pass to the limit by the dominated convergence theorem. The same proof works if the absolute drift coefficients and the trace of the diffusivity have integrable time integrals along the path on every finite horizon. Local hypotheses alone give a local martingale conclusion.