By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon:
"$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$
where $h = h(x)$ is a truncation function, $B = (B_t)_{0 \leq t \leq T}$ is a predictable process of bounded variation, $H^c = (H^c_ t )_{0 \leq t \leq T}$ is the continuous martingale part of $H$ with predictable quadratic characteristic $\langle H^c \rangle = C$, and $\nu$ is the predictable compensator of the random measure of jumps $μ$ of $H$. Here $W \ast μ$ denotes the integral process of $W$ with respect to $μ$, and $W \ast (μ − \nu )$ denotes the stochastic integral of $W$ with respect to the compensated random measure $μ − \nu$."
I want to find the stochastic differential for $H_t$, $dH_t$? Is it possible for me to use a Feynman-Kac type formula to get the PDE for the characteristic function of $H_t$?
Any help will be appreciated! Cheers :)
For Feynman-Kac the answer in the general case is clearly no as PDE representations exist only in the Markovian case and as you may know general semi martingales are not always Markovian.
The general question is hard to answer (at lest for me) I have taken a look at Jacod and Shiryaev 's book and it is really quite technical and the existence of stochastic integral with respect to random measure is not a constructive one but rather an argument of the type 'there exist by theorem x.y.z a process that make the difference of two complicated expression a local martingale with the needed properties that we will call a stochastic integral". So it is hard to get an intuition in this case.
Anyway I think that the scope of application the paper is mainly Lévy processes so you should keep this in mind as those processes might have Feynman-Kac representation in terms of PIDE.
Best regards