Lévy–Khintchine formula

Every real Lévy process has a unique triplet $(a,b,K)$ with $b\ge 0$ and $\int (1\wedge x^2)K(dx)<\infty$ 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 ${\sigma }^2\ge 0$.

Given a Brownian motion $B$ and an independent Poisson random measure $N$ with intensity $dtK(dx)$, a Lévy process with triplet $(a,b,K)$ is

For the convention in the Lévy–Khintchine formula, a Lévy process has almost surely differentiable paths exactly when $b=0$ and $K=0$, and continuous paths exactly when $K=0$. It is integrable exactly when ${\int }_{|x|>1}|x|K(dx)<\infty$, and it has a finite second moment exactly when $\int x^{2}K(dx)<\infty$.