 Setup:
-------


Suppose $X\_t$ solves the SDE
$$
dX\_t = \mu(t,X\_t)dt +\sigma(t,X\_t)dZ\_t,
$$
where $Z\_t$ is a Lévy process on $\mathbb{R}^d$, $g(t,s,x):[0,T]\times[0,1]\times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is $C^{1,2,2}$ and $S\_t$ is an SDE solving
$$
dS\_t = a(t,S\_t,X\_t)dt + b(t,S\_t,X\_t)dY\_t,
$$
where $Y\_t$ is a Lévy process as well. 


Question:
---------


Is it possible to obtain an SDE for:
$$
\mathbb{E}[g(t,S\_t,X\_t)|\sigma(X\_s)\_{0\leq s\leq t}]?
$$
That is the projection of $g(t,S\_t,X\_t)$ onto the information only known throught $X\_t$ up to time $t$?


Idea:
-----


I was considering some type of filtering approach, but I'm not certain how to go about this.



SDE for conditioning on subfiltration
You can obtain an S*P*DE (or, rather, an S*I*DE - stochastic integro-differential equation) for the conditional density of $S$ given the history of $X$. From here you can calculate the quantity you're looking for easily.


In the continuous case (i.e. $Z$ and $Y$ are Wiener processes), there are plenty of references, look for Zakai's equation. When you have jumps, the main reference I know of is B. Grigelionis and R. Mikulevicius: Nonlinear filtering equations for stochastic processes with jumps. 


