Lévy Processes: (Series Number 121 Cambridge Tracts in Mathematics)

This is an up-to-date and 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.

Table of Contents:
Preliminaries; 1. Lévy 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 Lévy process; 6. Fluctuation theory; 7. Lévy processes with no positive jumps; 8. Stable processes and the scaling property; Bibliography; Glossary; Index.

Review :
'At last! For many years, the stochastic community has awaited the publication of a textbook on Levy processes ... Here it is ... The author presents us with a thorough, concise and very readable account ... It also forms the ideal source on which to base a (post) graduate course on the subject ... the book is a gem and belongs on every probabilist's bookshelves.' P. A. L. Embrechts, Short Book Reviews ' ... an up-to-date and comprehensive account of the theory of Levy processes.' L'Enseignement Mathematique 'I think this is THE book on the subject, rather than A book on it. The text is clearly written, and very well organised. The subject-matter is mainstream probability, so will always be topical. A book on these lines has been long overdue ... I see the book as being both accessible enough for rookies to learn the subject from it, and authoritative enough for specialists to use for reference.' Professor Nick Bingham, Burbeck College 'This book would form an excellent text for a graduate course intended for probabilists. But it also provides a quick way of accessing and understanding fifty years of the work of many researchers. I strongly approve of both the choice of material and the pleasant manner in which it has been assembled.' S. J. Taylor, Bulletin of the London Mathematical Society ' ... this concise book promises to be the standard reference for students and researchers concerned with this field.' International Mathematical News