Carré du champ operator is a quadratic variation

Let $$X_t$$ be a real valued Markov process (starting at $$x$$) with generator $$L$$. Let $$\Gamma(f)$$ denote Carré du champ operator i.e. $$L(f^2) - 2f \cdot L (f)$$. As far as I know under suitable regularity assumptions: $$M_t^2 - \int_{0}^{t} \Gamma(f)(X_s) ds$$ is a martingale, where $$M_t := f(X_t) - f(x) - \int_{0}^{t}Lf(X_s)ds$$ (which is also a martingale by a fairly standard result) . I'm looking for sources which elaborate a little bit on this fact. So far I've only seen proof for diffusion processes. Any help greatly welcomed.