Lévy processes. (English) Zbl 0861.60003

Bertoin, Jean

Cambridge Tracts in Mathematics. 121. Cambridge: Cambridge Univ. Press. x, 265 p. (1996).

This monograph is devoted to the study of stochastic processes with independent and stationary increments in the Euclidean framework. These are called Lévy processes and can be thought of as random walks in continuous time. Their study has been developed over recent years and has many applications in such areas as queues, dams, mathematical finance and risk estimation. The book is divided into 9 chapters. The first one introduces the notation and reviews some elementary material on infinitely divisible laws, Poisson processes, martingales, Brownian motion, and regularly varying functions. The core of the theory of Lévy processes in connection with the Markov property and the related potential theory is developed in the second and third chapters. The fourth chapter is devoted to subordinators, which form the class of increasing Lévy processes, insisting on the properties of their sample paths. Subordinators also play a key part in the fifth chapter, where Itô’s theory of the excursions of a Markov process away from a point is introduced, and in the sixth chapter, where the local times of Lévy processes are investigated. The fluctuation theory is presented in the seventh chapter, following the Greenwood-Pitman approach based on excursion theory. The eighth chapter is devoted to Lévy processes with no positive jumps, for which fluctuation theory becomes remarkably simple. Finally, several consequences of the scaling property of stable processes are presented in the last chapter. Each chapter ends with exercises, which provide additional information, and with comments, where credits and further references are given.
Using the powerful interplay between the probabilistic structure and analytic tools (especially Fourier and Laplace transforms), the author has succeeded in giving a quick and concise treatment of the core theory, with the infimum of technical requirements. The book is as self-contained as possible, the prerequisites being limited to standard notions in probability and Fourier analysis. In the reviewer’s opinion the present monograph is destined to become the standard reference on the subject.

Reviewer: M.Iosifescu (Bucureşti)

MSC:

60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60J99	Markov processes
60G10	Stationary stochastic processes
60G17	Sample path properties
60J55	Local time and additive functionals

Keywords:

core theory; independent and stationary increments; Lévy processes; mathematical finance; risk estimation; Greenwood-Pitman approach; excursion theory