For $B$ a Brownian motion and the SDE $$dX_t=f(X_t)dt+g(X_t)dB_t,\qquad X_0=x$$ where $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ are Lipschitz.
As the title says, I need to find the function $\phi\in C^2(\mathbb{R},\mathbb{R})$ for which this $W_t=\phi(X_t)$ is a local martingale.
Now, what I believe but I'm not completely sure of, is to use that $$M_t^\phi=\phi(X_t)-\phi(X_0)-\int_0^t(L_s\phi(X_s))ds$$ is a local martingale. I believe this is called the (local) martingale problem. Is this the right approach? If so, how am I able to use this result to find my function of $\phi$?