Let $\{N(t):t\geqslant0\}$ be a Poisson process with intensity $\lambda$ and consider the compensated process $M(t) = N(t) - \lambda t$. It is well-known that $M(t)$ is a martingale. Is it analytically tractable to compute the distribution $\mathbb P(M(t)\leqslant s)$ for $s\in\mathbb R$? This is equivalent to $\mathbb P(N(t)\leqslant \lambda t+s)$, and the Poisson distribution does not have a nice closed form, so I have not had much success.
If not, can we reasonably approximate its distribution, perhaps for large $t$?