= Solution
Take $f\in C_b^2$. Boundedness of $a,b$ and the derivatives gives a finite bound $\|Lf\|_\infty$. On $0\leq t\leq T$ the assumed local <martingale> satisfies
$$
\left|f(X_t)-f(X_0)-\int_0^tLf(X_s)\,ds\right|\leq2\|f\|_\infty+T\|Lf\|_\infty.
$$
It is therefore a bounded local <martingale> on this time interval. To check the true <martingale> property directly, stop at a localizing sequence, apply its conditional <martingale> identity, and pass to the limit by <dominated convergence theorem> using this deterministic bound. Since $T$ is arbitrary,
$$
\boxed{X\text{ solves the diffusion martingale problem }\mathbf M(a,b).}
$$
The hypothesis for every $C^2$ test function contains the required $C_b^2$ class. No integrability assumption on the unbounded test functions is needed.
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