Show that $((N_t-t)^2-t)_{t \geq 0}$ is a martingale for a Poisson process $(N_t)_{t \geq 0}$

Question

I am asked to show that if $N$ is a poisson process of intensity $1$, then:

1. $X_t=N_t-t$ is a martingale.
2. $X_t^2-t$ is a martingale.

I have done the first part easily, using independence of increments. The second part I am having trouble with. Can anyone help?

Answer 1

Hint: Write $$X_t^2 = \bigg( \big[ (N_t-N_s)-(t-s) \big]+ \big[ N_s-s \big] \bigg)^2$$ for $s \leq t$ and expand the square. In order to calculate the conditional expectation $\mathbb{E}(X_t^2 \mid \mathcal{F}_s)$, consider the terms separately and use that

• $(N_t-N_s)-(t-s)$ is independent of $\mathcal{F}_s$
• $N_t-N_s \sim N_{t-s}$