Solution to the martingale problem for continuous-time Markov process $(X_t)$ with generator $\mathcal{L}$ is
$$M_t=f(X_t)-f(X_0)-\int_0^t\mathcal{L}f(X_s)ds,$$
given any $f$.
What I want to show is that the quadratic variation of $M_t$ is
$$\langle M\rangle_t = \int_0^t\{\mathcal{L}(f^2)-2f\mathcal{L}f\}(X_s)ds,$$
which is equivalent to showing that
$$ N_t =2\int_0^tf(X_s)\mathcal{L}f(X_s)ds - 2f(X_t)\int_0^t\mathcal{L}f(X_s)ds + \left(\int_0^t\mathcal{L}f(X_s)ds \right)^2 $$
is a martingale.
Any ideas or comments are appreciated!! Thank you.