Let $B$ be Brownian motion and let $N(ds,dx)$ be an independent <Poisson random measure> with intensity $ds\,K(dx)$. Writing $\widetilde N=N-ds\,K(dx)$ for its compensated version, the <Lévy–Itô decomposition> constructs
$$
\boxed{
X_t=at+\sqrt b B_t
+\int_0^t\!\int_{|x|\leq1}x\,\widetilde N(ds,dx)
+\int_0^t\!\int_{|x|>1}x\,N(ds,dx).}
$$
The four terms are independent drift, Gaussian, compensated small-jump and compound-Poisson large-jump components.

Solution

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

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A Poisson random measure $N$ with intensity measure $\mu$ assigns independent Poisson random variables to disjoint measurable sets, with $N(A)$ having mean $\mu(A)$. Its compensated version is $\widetilde N=N-\mu$ on sets of finite intensity.


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