I have asked a similar question involving some finance background some time ago here [math.stackexchange](https://math.stackexchange.com/questions/137011/futures-pricing-and-futures-price-process-under-the-real-world-measure), 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.



Solving an Ornstein-Uhlenbeck-like SDE $y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t]$
In the level of generality you have stated, I have no ideas. But in some cases this can be reformulated as a type of backward SDE, as studied in section 2.2 of <http://arxiv.org/pdf/0801.3505.pdf> in the case that $X$ and $H$ are continuous. It is no simple problem, but a couple of special cases seem approachable.


The simplest case is if $\int H\_- dX$ is a martingale. Then $y(t,T) = H\_t$ is a solution. An example if $X$ is a square integrable martingale and $H$ satisfies $\mathbb{E}[\int H^2\_- d[X,X]] < \infty$.


Somewhat more generally, suppose $X$ is a continuous semimartingale, so $X = M + A$ for unique continuous local martingale $M$ and continuous FV process $A$. Assume moreover that $M$ is a true martingale and that your filtration is generated by a Brownian motion $W$. Let's look for a solution $y(\cdot,T)$ among the class of continuous processes $Y$ such that $\int Y dM$ is a martingale. For such $Y$ the equation becomes


$Y\_t = H\_t + \mathbb{E}[\int\_t^TY\_sdAs | \mathcal{F}\_t]$.


To make this look more like a standard BSDE, note that if such a $Y$ exists then $N\_t = \mathbb{E}[\int\_0^TY\_sdAs | \mathcal{F}\_t]$ is a martingale and


$dY\_t = dH\_t - Y\_tdA\_t + dN\_t$, with $Y\_T = H\_T$.


Now forget about the constraint that $\int Y dM$ is a martingale, and let's look for a pair of processes $(Y,N)$, with $N$ a martingale, satisfying this equation. This is a special case of the problem studied in the aforementioned paper: see equation (2.19) with $\xi = H\_T$, $J = -H$, $g \equiv 0$, and $f(s,y) = y$. Theorem 2.2 of said paper will guarantee you the existence of a unique process $Y \in \mathbb{S}^p$ satisfying this equation, as long as $H \in \mathbb{S}^p$ and there exist two continuous BMO martingales $N\_1$ and $N\_2$ with $\langle N\_1, N\_2 \rangle = A$. Here $\mathbb{S}^p$ is the set of continuous adapted processes whose supremum has finite $p^{th}$ moment. Now if it happens that $\int Z dM$ is a martingale whenever $Z \in \mathbb{S}^p$, then this $Y$ uniquely solves your equation.


