Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 6 / c / Solution

Let $B$ be Brownian motion and let $N(ds,dx)$ be an independent Poisson random measure with intensity $dsK(dx)$. Writing $\tilde{N}=N-dsK(dx)$ for its compensated version, the Lévy–Itô decomposition constructsThe four terms are independent drift, Gaussian, compensated small-jump and compound-Poisson large-jump components.