# What is the stochastic differential of a general semimartingale?

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 $H_t$?

• 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. Nov 10 '13 at 2:36

If I am not mistaken, the Feynman-Kac formula is related to the Kolmogorov's backward equation, so I would expect it to be available only for Markov processes. Diffusions are usually of Markovian type, in contrast to general Ito processes or more to say, general semimartinagales. Intuitively, the PDE/PIDE/... will describe the dynamics of distribution/expectation in time based on the current spatial structure: $$\frac{\partial f}{\partial t} = Af \tag{1}$$ where $A$ is a suitable linear operator (differential in case of diffusion, integro-differential for jump-diffusions). The dependence of the type $(1)$ certainly hints upon the Markovian structure, so I would not expect useful F-K formulas to be available for general semiartingales. I say useful, since you can always include the whole history of the process as a state to try to express this as a Markov process, but over such an enlarged state space, I don't think F-K would be of any use even though it would be available.
Regarding the stochastic differential, as with usual differential you are trying to decompose one function dynamics w.r.t. another. That is, $dH_t$ is a stochastic differential on its own, and it can be used as an integrator per theory of semimartingale stochastic integration. Another question is whether $df(H_t)$ can be expressed in terms of $H_t$. This is indeed true, see the general Ito formula in Theorem 2.7.1 here.