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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?

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What about having the integrand to be adapted to the filtration of the integrator? If you are able to find a nice filtration, then remember that any Poisson process is independent of Brownian motion (see e.g. in Shreve, Stochastic Calculus for Finance II, Chapter 11) –  Ilya Apr 3 '12 at 12:09
    
This is in a financial setting with a completed, right continuous filtration generated by jump times/sizes of $N_t$ and $S_t$ - so satisfying the usual assumptions. –  user13655 Apr 3 '12 at 12:46
    
so what about using the independence of $N_t$ and $S_t$? –  Ilya Apr 3 '12 at 13:03
    
Sorry, I don't quite see how. A short hint would be appreciated. –  user13655 Apr 3 '12 at 13:30
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Well, I think it goes like $$ \mathsf E\left[\int_0^T N_{t}dS_t\right] = \mathsf E\left[\mu\int\limits_0^TN_{t}S_tdt\right]+\mathsf E\left[\sigma\int\limits_0^TN_{t}S_tdB_t\right] = \mu\int\limits_0^T\mathsf E[N_t]\mathsf E[S_t]dt $$ where 1. you can eliminate $t-\mapsto t$ since $S_t$ does not have jumps 2. we use that $N_t$ and $S_t$ are independent and 3. expectations in the latter integral have closed forms. For details use the book I referred to above. I don't have it at hand - so the answer is not rigorous, but it is OK for the hint, I hope. –  Ilya Apr 3 '12 at 13:43

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