How to prove that $(N-\lambda t)^2 - \lambda t$ is a martingale where $N$ is Poisson process

Question

Let $N$ be a Poisson process with a parameter $λ>0$. Can anyone help to show that $(N-\lambda t)^2 - \lambda t$ is a martingale?

Answer 1

For $0\leqslant s \leqslant t$, we have \begin{align*} &\mathbb E[(N_t-\lambda t)^2 -\lambda t |\mathcal F_s]=\mathbb E[(N_s-\lambda s + N_t-N_s -\lambda(t-s))^2 | \mathcal F_s]-\lambda t\\ &=\mathbb E[(N_s-\lambda s)^2 + 2(N_s-\lambda s)(N_t-N_s -\lambda(t-s)) + (N_t-N_s -\lambda(t-s))^2 | \mathcal F_s] -\lambda t\\ &=(N_s -\lambda s)^2 + 2\mathbb E[(N_s-\lambda s)(N_t-N_s -\lambda(t-s))|\mathcal F_s]+ \mathbb E(N_t-N_s -\lambda(t-s))^2 -\lambda t\\ &=(N_s -\lambda s)^2 + 0 + \lambda(t-s)-\lambda t=(N_s -\lambda s)^2 -\lambda s. \end{align*} This implies $((N_t -\lambda t)^2 -\lambda t)_{t\geqslant 0}$ is a martingale.