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 a bounded-domain diffusion with almost surely finite exit time ${\tau }_D$, bounded $L$-harmonic functions continuous up to the boundary satisfy $u(x)={\mathbb{E}}_{x}u(X_{{\tau }_D})$. Stop the test-function martingale and use bounded convergence. Identical boundary values therefore imply identical harmonic functions. Uniform ellipticity and bounded SDE coefficients ensure finite exit by the projection argument.

// Target: probability-and-statistics.bigb

For an SDE with bounded drift and diffusion coefficients and ${\sum }_k⟨\xi ,{\sigma }_k(x){⟩}^2\ge \lambda |\xi {|}^2$, each fixed nonzero scalar projection of a solution is unbounded in absolute value almost surely. Rescale a slab to $(-1,1)$; its projected drift is controlled by its uniformly positive bracket density. The exit with drift controlled by quadratic variation then forces exit from every slab in a countable exhaustion.

// Target: probability-and-statistics.bigb

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.