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By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon's "Analysis of Fourier Transform Valuation Formulas and Applications", on page 3:

$$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$$

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 an SDE for $H_t$.

What is the stochastic differential for $H_t$, $dH_t$?

Is it possible to use a Feynman-Kac type formula to get the PDE for the characteristic function of $H_t$?

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You'll only get a PDE if your underlying process $H$ is diffusion. If you add a Poisson or Levy process, you will get a PIDE, which has an integral term in it: this results from the non-localized nature of the jumps that $H$ can take. The Levy-Khinchin formula calculates the characteristic function in terms of $H$'s characteristics, so that might be what you're looking for. But, if $H$ is a general semi-martingale like you described, I believe the characteristic function will satisfy a stochastic PDE, which is studied, but not that widely. –  quasi Nov 10 '13 at 2:36
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