 Solution

Take $f\in C_b^2$. Boundedness of $a,b$ and the derivatives gives a finite bound $\Vert Lf{\Vert }_{\infty }$. On $0\le t\le T$ the assumed local martingale satisfiesIt 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,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.