I have a compound Poisson process $(X_t)$, with jump distribution $F$, which assigns mass only to $(0,\infty)$. In my working I have an expression of the following form:
$$ \mathbb{E} \int_0^{\tau} g(t,X_t) dt $$
where $\tau$ is a stopping time relative to the natural filtration of $(X_t)$, and $g$ is some suitable function (actually in my case it is non-negative and the mapping $ t \mapsto g(t,X_t)$ is almost surely increasing).
(In fact, my case is even simpler: $g(t,X_t) = t f(X_t)$, for some increasing, non-negative function $f $).
I am wondering, are there standard methods for analysing this expression, and getting answers in terms of $F$? This seems like a fairly natural set up to me, but I don't know what techniques are appropriate.
Many thanks for your help.