## 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.