I try to prove that the $N^{2}(t)-t$ is martingale, where $N(t)$ is a poisson process, i.e $N(t)\sim Pois(\lambda t)$ or $N(t)-N(s)\sim Pois(\lambda(t-s))$ for $s<t$.
Hence, I have to prove that $\mathbb{E}[N^{2}(t)-\lambda t|F_{s}]= N^{2}(s)-\lambda s$
Equivalently, I try to prove that $\mathbb{E}[N^{2}(t)-N^{2}(s)-\lambda(t-s)|F_{s}]=0$
$\mathbb{E}[(N(s)+(N(t)-N(s))^{2}-N^{2}(s)|F_{s}]-\lambda(t-s)=$
$\mathbb{E}[N^{2}(s)+(N(t)-N(s))^{2}+2N(s)(N(t)-N(s))-N^{2}(s)|F_{s}]-\lambda(t-s)=$
$\mathbb{E}[(N(t)-N(s))^{2}+2N(s)(N(t)-N(s))|F_{s}]-\lambda(t-s)$
The expected value of $(N(t)-N(s))^{2}$ is $\lambda(t-s)+(\lambda(t-s))^{2}$.
Thus,
$= \lambda(t-s)+\lambda^{2}(t-s)^{2}+2\lambda^{2} s(t-s) -\lambda(t-s)= \lambda^{2}(t-s)(t+s)\neq0$
Which is clearly not a martingale.
Is there something on my calculations that is wrong ??