Lévy–Khintchine formula

Every real Lévy process has a unique triplet $(a,b,K)$, with $a\in \mathbb{R}$, $b\ge 0$ and a Lévy measure $K$ satisfying $\int (1\wedge x^2)K(dx)<\infty$ such that

If a Lévy measure satisfies ${\int }_{|y|\le 1}|y|K(dy)<\infty$, the jump integral in the Lévy–Khintchine formula can be written without compensation: $\mathbb{E}{e}^{iu{X}_t}=e^{t\psi (u)}$, with the displayed exponent. Its integral is absolutely convergent because $|e^{iuy}-1|\le |u||y|$ near zero and is at most $2$ elsewhere. If the compensated formula uses drift $\gamma$, the uncompensated drift is $b=\gamma -{\int }_{|y|\le 1}yK(dy)$. The Gaussian coefficient is still ${\sigma }^2\ge 0$.

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.

For the small-jump cutoff convention in the Lévy-Khintchine formula, let $(a,b,K)$ be the triplet of a real Lévy process. Its paths are continuous almost surely exactly when $K=0$. They are everywhere differentiable almost surely exactly when $K=0$ and $b=0$, giving the deterministic path $X_t=at$. A compound Poisson path can have a derivative between its jumps without being an everywhere differentiable path.

finite variation on compact time intervals has the different criterionUnder these conditions the uncompensated Lévy-Khintchine formula gives a drift plus an absolutely summable small-jump process and finitely many large jumps. A nonzero Brownian component precludes finite variation, while an infinite absolute small-jump sum already forces infinite variation.

Integrability requires only the large-jump first moment:The small compensated jumps are square integrable and hence integrable. When this condition holds, $\mathbb{E}{X}_t=t[a+{\int }_{|x|>1}xK(dx)]$; the small-jump compensation convention is why no small-jump mean appears.

Likewise a finite second moment is equivalent to $\int x^{2}K(dx)<\infty$, and thenThe Lévy-Itô decomposition proves these moment statements by independent Brownian, compensated-small-jump and compound-Poisson-large-jump parts. The large-jump one-event case also shows why the tail moment conditions are necessary.