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osprey

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

[itex] \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[/itex]?

(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

[itex] \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[/itex]?

(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

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