# Stochastic differential for general semimartingale

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 :)

• @ Chaturi Bhaskaran : When you say in "Eberlein, Glau and Papapantoleon" could you please be more accurate on the reference you give ? Best regards. – TheBridge Nov 6 '13 at 9:45
• @TheBridge Yes, sorry! The paper is called "ANALYSIS OF FOURIER TRANSFORM VALUATION FORMULAS AND APPLICATIONS" linked here stochastik.uni-freiburg.de/~eberlein/papers/analysis_090911.pdf, the representation I am referring to is on page 3. – Chaturi Bhaskaran Nov 6 '13 at 11:15
• @ Chaturi Bhaskaran : Ok Thank's. – TheBridge Nov 6 '13 at 12:21