This 1996 book is a comprehensive account of the theory of Levy processes. This branch of modern probability theory has been developed over recent years and has many applications in such areas as queues, mathematical finance and risk estimation. Professor Bertoin has used the powerful interplay between the probabilistic structure (independence and stationarity of the increments) and analytic tools (especially Fourier and Laplace transforms) to give a quick and concise treatment of the core theory, with the minimum of technical requirements. Special properties of subordinators are developed and then appear as key features in the study of the local times of real-valued Levy processes and in fluctuation theory. Levy processes with no positive jumps receive special attention, as do stable processes. In sum, this will become the standard reference on the subject for all working probability theorists.
Preliminaries; 1. Levy processes as Markov processes; 2. Elements of potential theory; 3. Subordinators; 4. Local time and excursions of a Markov process; 5. Local times of a Levy process; 6. Fluctuation theory; 7. Levy processes with no positive jumps; 8. Stable processes and the scaling property; Bibliography; Glossary; Index.