How can I prove it is a martingale when there is a jump process

Question

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.)

Answer 1

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.