 Lévy–Itô decomposition

The Lévy-Itô decomposition builds a real Lévy process from drift, continuous Gaussian noise and Poisson jumps. For the triplet $(a,b,K)$ in the Lévy-Khintchine formula with small-jump cutoff one, take independent standard Brownian motion $B$ and Poisson random measure $N$ of intensity $dsK(dx)$. ThenHere $b\ge 0$, $K$ is a Lévy measure and $\tilde{N}=N-dsK(dx)$ 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 $\epsilon >0$, let $J_t^{\epsilon }={\int }_0^t{\int }_{\epsilon <|x|\le 1}x\tilde{N}(ds,dx)$. These are square-integrable martingales. The compensated Poisson isometry givesas $0<\delta <\epsilon \downarrow 0$. Doob's L2 maximal inequality also controls the supremum on $[0,T]$, so the limit exists locally uniformly in probability, indeed in the L2 norm of that supremum. This uses exactly the small-jump second-moment condition of a Lévy measure.^[ref]

The components have independent increments; multiplying their 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.