Lévy processes revisited

I’ve been thinking about Lévy processes, a topic that I mentioned once before. A Lévy process is a generalization of both Brownian motion and a Poisson process. Brownian motion and Poisson processes are both continuous-time stochastic processes but have very different behavior. A Brownian motion follows a very jagged path that is almost always continuous. A Poisson process stays at one place for a long time, and then suddenly jumps to a new place. What they have in common is that changes over two disjoint time intervals are independent of each other, and if the two time intervals are the same the changes have the exact same distribution.

Lévy processes include generalizations such as various combinations of Brownian motions and a Poisson process, but they also include more exotic possibilities. The sample path of a combination of a Brownian motion and a Poisson process will almost always have only finite number of discontinuities. In general, Lévy processes can generate sample paths with infinitely many jump discontinuities in almost every interval. Over a finite time horizon, this can give rise to fait-tailed distributions such as the Cauchy distribution.