I know that $M_t=N_t-\lambda t$ is a martingale for $N_t$ a rate $\lambda$ poisson process and that for a brownian motion, $B_t^2-t$ is a martingale. I'm wondering, is there something similar for $M_t^2$? Like some function of t, say $a_t$, such that $M_t^2-a_t$ is a martingale. If so how would you go about finding one?
Any help is appreciated!