For a Markov process $X_{t}$ with $X_{0}=x$ and infinitesimal generator $\mathcal{L}$, we have the following martingale \begin{align*} M_{t} = f(X_{t})-f(x)-\int_{0}^{t}\mathcal{L}f(X_{s})ds \end{align*} for any suitable test function $f$. Without knowing more about $X_{t}$ (in particular, not assuming it is continuous), what is the predictable quadratic variation of $M_{t}$? That is, what is the unique increasing process $A_{t}$ such that $M_{t}^{2}-A_{t}$ is a martingale?
I've been told that $A_{t} = \int_{0}^{t}\mathcal{L}(f^{2})(X_{s})-2\mathcal{L}{f}(X_{s})ds$, but I don't know how to show this.
Related questions:
- Is there are name for the martingale $M_{t}$?
- What is the moment generating function of $M_{t}$?