Poisson exponential martingale ($M_t^{\theta }=\exp (\theta N_t-t(e^{\theta }-1))$)

For a unit-rate Poisson process and real $\theta$, the displayed process is a nonnegative martingale with initial value one. Independent increments give $\mathbb{E}{e}^{\theta (N_t-N_s)}=\exp ((t-s)(e^{\theta }-1))$, proving its conditional mean property. The Ville inequality turns this exponential moment into a maximal tail bound.