1 - Lévy processes

Summary

Summary Section 1.1 is a review of basic measure and probability theory. In Section 1.2, we meet the key concepts of the infinite divisibility of random variables and of probability distributions, which underly the whole subject. Important examples are the Gaussian, Poisson and stable distributions. The celebrated Lévy–Khintchine formula classifies the set of all infinitely divisible probability distributions by means of a canonical form for the characteristic function. Lévy processes are introduced in Section 1.3. These are essentially stochastic processes with stationary and independent increments. Each random variable within the process is infinitely divisible, and hence its distribution is determined by the Lévy–Khintchine formula. Important examples are Brownian motion, Poisson and compound Poisson processes, stable processes and subordinators. Section 1.4 clarifies the relationship between Lévy processes, infinite divisibility and weakly continuous convolution semigroups of probability measures. Finally, in Section 1.5, we briefly survey recurrence and transience, Wiener–Hopf factorisation, local times for Lévy processes, regular variation and subexponentiality.

Review of measure and probability

The aim of this section is to give a brief resumé of key notions of measure theory and probability that will be used extensively throughout the book and to fix some notation and terminology once and for all. I emphasise that reading this section is no substitute for a systematic study of the fundamentals from books, such as Billingsley, Itô,Ash and Doléns-Dade, Rosenthal, Dudley or, for measure theory without probability, Cohn.