= Lévy–Itô decomposition
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The <Lévy–Itô decomposition>[Lévy-Itô decomposition] builds a real <Lévy process>[Lévy process] from drift, continuous <Gaussian noise> and Poisson jumps. For the triplet $(a,b,K)$ in the <Lévy–Khintchine formula>[Lévy-Khintchine formula] with small-jump cutoff one, take <independent> standard <Brownian motion> $B$ and <Poisson random measure> $N$ of intensity $ds\,K(dx)$. Then
$$
X_t=at+\sqrt b\,B_t
+\int_0^t\int_{|x|\leq1}x\,\widetilde N(ds,dx)
+\int_0^t\int_{|x|>1}x\,N(ds,dx).
$$
Here $b\geq0$, $K$ is a <Lévy measure>[Lévy measure] and $\widetilde N=N-ds\,K(dx)$ denotes compensation. The large-jump term has finitely many jumps on each bounded time <interval> because $K(|x|>1)<\infty$.
The small-jump <integral> is defined by limits, rather than by subtracting two potentially infinite sums. For $\varepsilon>0$, let $J_t^{\varepsilon}=\int_0^t\int_{\varepsilon<|x|\leq1}x\,\widetilde N(ds,dx)$. These are <square-integrable martingale>[square-integrable martingales]. The <compensated Poisson integral>[compensated Poisson isometry] gives
$$
\mathbb E|J_T^{\delta}-J_T^{\varepsilon}|^2
=T\int_{\delta<|x|\leq\varepsilon}x^2K(dx)\longrightarrow0
$$
as $0<\delta<\varepsilon\downarrow0$. <Doob L2 maximal inequality>[Doob's L2 maximal inequality] also controls the <supremum> on $[0,T]$, so the limit exists locally <uniform convergence on compacts in probability>[uniformly in probability], indeed in the <L2 norm> of that <supremum>. This uses exactly the small-jump second-moment condition of a <Lévy measure>[Lévy measure].<Lectures on the Poisson Process by Günter Last and others>[Chapter 12: compensated Poisson integrals and their isometry]{ref}
The components have <independent increments>; multiplying their <characteristic function>[characteristic functions] recovers the Lévy-Khintchine exponent. The decomposition separates continuous fluctuations from large jumps and possibly infinitely many small jumps. A finite small-jump <Second moment> does not require a finite first absolute jump <moment>, which explains why compensation and its limiting construction are essential.
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