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

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

For $g\ge 0$, the Tonelli theorem and conditioning on $N_t$ giveSince $\mathbb{E}[N_t]=\lambda t$,with both sides allowed to be $+\infty$.

A martingale must be integrable. On the event $\{N_t=1\}$, $X_t=g(Y_1)$, so integrability of $X_t$ forcesFor $s<t$, independent increments giveTherefore the necessary and sufficient condition is

The joint process is a two-dimensional Compound Poisson process whose Lévy measure isTwo coordinates of a Lévy process are independent exactly when its Lévy measure charges only the coordinate axes and its Gaussian covariance has no cross term. Here there is no Gaussian part, so independence is equivalent toSince $g_1{g}_2$ is continuous, this is equivalent to pointwise vanishing. Conversely, when the product vanishes, the mark sets where $g_1$ and $g_2$ are nonzero are disjoint; independent thinning of the Poisson random measure gives independent coordinate processes. Hence