Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 6 / a / Solution

The count $N_t=M(0,t]$ is a rate-$\lambda$ Poisson process. Over a time interval $(s,t]$, the incrementdepends only on the Poisson points and marks in that interval. Disjoint intervals give independent increments, and the distribution depends only on $t-s$. The paths are càdlàg step functions, $X_0=0$, andThus $X$ is stochastically continuous and