# SDE for conditioning on subfiltration

## 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.

You can obtain an SPDE (or, rather, an SIDE - 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.