A <Poisson process> of rate $\lambda>0$ is a counting process with $N_0=0$, <independent> stationary increments, and $N_t-N_s\sim\operatorname{Poisson}(\lambda(t-s))$ for $0\le s<t$. It has nondecreasing right-<continuous> integer-valued paths with <counting process>[unit jumps] almost surely. The <independent> increment requirement distinguishes the process from merely having Poisson marginals.

Solution

Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / M / M4 / b / Solution

 Solution (source code)