I'm sorry if this question may have been asked before but given the dearth of references, I doubt it.
Now, here's my problem. I have this continuous function $f(t,w,x)$ which is dependent on time t, Levy process $w$, and continuous time Markov chain $x$. I want to derive the corresponding Ito's Lemma. I have no problem handling Levy processes because there's lot of references. My problem is with the derivative of $f$ with respect to the Markov chain $x$. I have found journal articles about it but they don't provide the detailed derivation. With respect to the Markov chain, they just provide this expression
$\frac{\partial f}{\partial x}=\sum_{j\neq i}q_{ij}[f(j)-f(i)]+[f(j)-f(i)]dM$
where $q_{ij}$ is the generator of the Markov chain for $i,j\in\mathcal{M}=\{1,2,\cdots,n\}$, $M$ is a martingale.
I have been patiently reading books and journals for like months now but can't still pin down the derivation of this chain rule for Markov chain. I would be extremely grateful if you can help me.
Thanks.