I have asked a similar question involving some finance background some time ago here math.stackexchange, however no really good answer came up. I was able to find a solution at least for a special case. Removing all unnecessary information, I try to solve the following problem.

Given a given a martingale $H_t$ and a semimartingale $X_t$, let $y(t,T)$ be the process solving

$ y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t] $

if a solution exists. I wouldn't even call this an SDE. I was able to come up with a solution in case of a deterministic $X_t$, but got stuck otherwise. It looks very similar to a OU-type SDE and looking at the solution of this kind of SDE, I thought

$y(t,T)=\mathbb{E}\left[ \mathcal{E}(X)_t\left( \frac{H_t}{\mathcal{E}(-X)_T}+\int_t^T\mathcal{E}(X)^{-1}_s(dH_s-d\langle H,X\rangle_s) \right)|\mathcal{F}_t\right]$

might work. This was some kind of educated guess, and for deterministic $X$ works fine (the integral vanishes). Also since $H$ is a martingale the integral simplifies. I'd like to solve for general $X$ (or at least an Ito-process).

Can standard SDE-Theory be applied here in any way? Is there a general method to solve such problems? Results about existence and uniqueness? I couldn't find anything in the literature. I'd be grateful for any hints.