The Lévy–Khinchin Formula
Every Lévy process is a drift, a diffusion coefficient, and a jump measure.
The idea
A Lévy process starts at $0$ and has independent increments whose law depends only on the elapsed time. Brownian motion is one, the Poisson process is another, and so is any independent sum of such pieces. The Lévy-Khinchin formula classifies the whole family by three ingredients: a drift $b$, a diffusion coefficient $\sigma^{2} \ge 0$, and a Lévy measure $\nu$ on $\mathbb{R} \setminus \{0\}$ recording how often jumps of each size occur.
The classification runs through the characteristic function $\mathbb{E}\big[e^{iuX}\big]$, a complex-valued function of a real $u$ that determines a distribution completely. The two defining properties force $\mathbb{E}\big[e^{iuX_{t}}\big] = e^{t\,\psi(u)}$, so a single function $\psi$, the characteristic exponent, carries the whole process, and classifying the family means classifying the possible $\psi$.
Theorem (Lévy-Khinchin formula).
The characteristic exponent of a Lévy process is a function of the form $\psi(u) = ibu - \tfrac{1}{2}\sigma^{2}u^{2} + \int_{\mathbb{R}\setminus\{0\}}\Big(e^{iux} - 1 - iux\,\mathbf{1}_{\{|x| < 1\}}\Big)\,\nu(dx),$ for a unique triple $(b, \sigma^{2}, \nu)$ with $b \in \mathbb{R}$, $\sigma^{2} \ge 0$, and $\nu$ a measure on $\mathbb{R} \setminus \{0\}$ satisfying $\int_{\mathbb{R}\setminus\{0\}} \min(1,\,x^{2})\,\nu(dx) < \infty.$ Conversely, every such triple is the triple of some Lévy process.
The three terms are the three motions available to such a process: steady drift, continuous fluctuation, and jumps. The subtraction inside the integral is not a fourth ingredient but a repair — without it the integral would diverge for a process making infinitely many small jumps — and the integrability condition says which measures $\nu$ are admissible.
Ways to work on it
- Walkthrough. Building the characteristic exponent from Brownian motion with drift and compound Poisson, then the compensation term.
- Practice. Translate between a Lévy process and its triple of drift, diffusion, and jump measure — in both directions.
- Hardest. When the formula's compensation term is genuinely required, and the variance of an assembled Lévy process.
Not sure where to start? Take the ten-question placement test.