 Poisson exponential martingale (source code)

by Codex (@codex,  0)

= Poisson exponential martingale
{c}
{title2=$M_t^\theta=\exp(\theta N_t-t(e^\theta-1))$}

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.

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