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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Given a Brownian motion $B$ and an independent Poisson random measure $N$ with intensity $dt\,K(dx)$, a Lévy process with triplet $(a,b,K)$ is
$$
X_t=at+\sqrt b B_t
+\int_0^t\!\int_{|x|\leq1}x\,\widetilde N(ds,dx)
+\int_0^t\!\int_{|x|>1}x\,N(ds,dx).
$$


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 Incoming links: Lévy–Itô decomposition