Let $N_t$ be a Poisson process with rate $\lambda$ and let $M_t=N_t-\lambda t$. I am then trying to find $$ \mathbb{E}\left[\int_0^tN_s\,\mathrm{d}M_s\right]. $$ I have tried applying the definition of an Îto integral to find that \begin{align*} \mathbb{E}\left[\int_0^tN_s\,\mathrm{d}M_s\right]&=\mathbb{E}\left[\sum_{j\geq0}N_{t_j}(M_{t_{j+1}}-M_{t_j})\right]\\ &=\sum_{j\geq0}\mathbb{E}[N_{t_j}]\mathbb{E}[N_{t_{j+1}}-N_{t_j}]-\lambda(t_{j+1}-t_j)\mathbb{E}[N_{t_j}]\\ &=\sum_{j\geq0}\lambda^2t_j(t_{j+1}-t_j)-\sum_{j\geq0}\lambda^2(t_{j+1}-t_j)t_j\\ &=\frac{1}{2}\lambda^2t^2-\frac{1}{2}\lambda^2t^2=0. \end{align*} However, from the way the next question is posed I feel this must not be right. The next question asks why $\int_0^tN_s\,\mathrm{d}M_s$ cannot be a martingale, which would be if its expectation depended on $t$ for example.
An even further question replaces $N_s$ with $N_{s^-}$, which I feel should result in the result above.
Hence, I feel I am missing an important step, can I not apply independence of increments in this way?
I have found this thread and viewed the mentioned work, but it didn't not clear things up for me.