I came across some nasty stochastic integral of which I'd like to calculate the expected value"
$\mathbb{E}[\int_0^T N_{t-} dS_t]$
where $N_t$ is a Poisson process and $S_t$ is, say, a geometric Brownian motion. The actual problem is more complex, but this makes it more accessible. All that came to my mind is conditioning on number of jumps, jump times of $N_t$ and so on. This is not very elegant and results in many numerical integrals. Does anyone know a way to attack this problem better?