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Let $N_t$ be a Possion process, $M_t=N_t - \lambda t$ we can easily show that $M_t$ is a martingale.

Now $\int_0^t\Phi_udM_u=.....=\sum_{i=1}^{N(t)}\Phi_{\tau_i}-\lambda\int_0^t \Phi(u)du $

$\tau_i$ is jumps time of the poisson process $N_t$.

The question is how to prove $\int_0^t\Phi_udM_u$ is a martingale?? You may give any extra conditions to $\Phi$ if needed. (I am more interested in the ideas to cope with this question instead of this question itself.)

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Hi I think you should add more inforamtions about properties of $\Phi$ if you want to get an answer. Best regards – TheBridge Jan 25 '13 at 16:21
If we want to easily show that $M_t$ is a martingale, we might better define it as $M_t=N_t-\lambda t$. – Did Jan 25 '13 at 21:07
Did: Thank you. That is a typos. – XXX11235 Jan 26 '13 at 16:33
TheBridge:I have editted the question. You may give any conditions if needed. Thanks. – XXX11235 Jan 26 '13 at 16:34
Any ideas to solve this question? – XXX11235 Jan 29 '13 at 19:16

We can prove rigorously that $(\tau_1, \tau_2....)$ given N(t)=n is uniformly distributed in $[0,t]^n$ space. Then the question can be solved easily.

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