Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 6

A Poisson random measure $N$ of intensity $\mu$ assigns to disjoint measurable sets independent random variables, andwhenever $\mu (A)<\infty$, with the usual countable-additivity requirement.

A Lévy process starts at zero almost surely, has stationary independent increments, and is stochastically continuous; one normally takes its càdlàg modification.

Independent Poisson processes have stationary independent increments, so their weighted sum does too and is stochastically continuous. Moreover,where

Let $N(ds,dy)$ be a Poisson random measure on $(0,\infty {)}^2$ with intensity $dsK(dy)$ and defineThe assumption $\int yK(dy)<\infty$ makes this integral finite on compact time intervals. The exponential formula for a Poisson random measure givesso $X$ is a Lévy process with exponent $\psi$.

Choose finite-valued measurable functions $q_n\ge 0$ which vanish off $[1/n,n]$ and satisfyThis is possible by truncation followed by approximation by simple functions. PutThe measure of the support of $q_n$ is finite, and $q_n$ takes finitely many values, so $X^n$ is a simple pure-jump Lévy process. Under this common coupling,