Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral
$$(H\star M)_t:=\int_0^t H_{s}dM_s=\int_0^t H_{s}dN_s-\lambda\int_0^t H_{s}ds\\
$$
is a martingale.
Here $N_t$ is a counting process defined as
$$
N_t=\sum_{n\ge 1}\mathbb{1}_{\{T_n\le t\}}=\sum_{n\ge 0}n\mathbb{1}_{\{T_n\le t<T_{n+1}\}}$$
where $(T_n,n\ge0)$ is a sequence of random times at which the jumps of $N_t$ happen. Moreover $\mathbb{P}(N_t=n)=e^{\lambda t}\frac{(\lambda t)^n}{n!}$.
The martingale $M_t$ is the compensated process associated to $N$, defined as $M_t=N_t-\lambda t$.
This proposition is interesting because it extends the martingale property of $M_t$ to the stochastic integration with respect to $M_t$ for bounded predictable integrands $H_t$.
I understand that this property can be proven by considering first simple integrands $H_t$ of the form $H_t=K_S \mathbb{1}_{]S,T]}(t)$, where S and T are two stopping times and $K_S$ is $\mathcal{F}$-measurable. Then, one can pass to the limit for general $H_t$.
But could anyone give a complete proof (or a reference to a complete proof) of these statements? This would also be very instructive as it is a very good example of working with Poisson Processes, Martingales, Stopping Times, Convergence Theorems and Stochastic Integration.
Moreover what would happen in the case $H_t$ is adapted, instead of predictable? Because this proposition seems to imply e.g. that the process $\int_0^t N_{s-}dM_s$ is a martingale, but the the process $\int_0^t N_{s}dM_s$ is not a martingale.
Best Answer
This might be helpful: Theorem 29 on page 173 of Protter's "Stochastic integration and differential equations":
Let $M$ be a local martingale and let $H\in \mathcal P$ be locally bounded. then the stochastic integral $H\cdot M$ is a local martingale.
I'm reading that book, but haven't got that far yet, will be interested to see if someone can provide a simple proof just for this special case in this post.