Why Is It Acceptable to Interchange Sum and Integral for Martingale Proof?

[osprey]

Let $M_t = \sum_{i=1}^\infty \frac{1}{i} (N_t^i - \lambda t)$ where $\{ N^i \}$ is a sequence of iid Poisson processes with intensity $\lambda$. It can be shown that the series converges in the $L^2$ sense. Why is it ok to write

$\int \left ( \sum_{i=1}^\infty \frac{1}{i} (N_t^i - \lambda t) \right ) dP = \sum_{i=1}^\infty \frac{1}{i} \int (N_t^i - \lambda t) dP$?

(I need to show that M is a martingale and to do so, I would like to interchange the sum and the integral, but I cannot find an argument that seems to work...)

Thank you in advance for your help! :-)

/O

 

[spamiam]

Hello osprey,

I'm not at all familiar with Poisson processes, but in general to exchange limits and integrals one falls back on the dominated convergence theorem: http://en.wikipedia.org/wiki/Dominated_convergence_theorem.