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**Lévy processes and infinitely divisible distributions.**
*(English)*
Zbl 0973.60001

Cambridge Studies in Advanced Mathematics. 68. Cambridge: Cambridge University Press. xii, 486 p. (1999).

This monograph presents a detailed treatment of most of the essential results on \(\mathbb{R}^d\)-valued Lévy processes, their time-inhomogeneous relatives, the additive processes, and their building brickes, the infinitely divisible distributions. Starting from the very beginning, all necessary notions are introduced, and all results are proven. Thus this book surely will be turn to a reference book on Lévy processes. It can be recommend as well for specialists on probability theory as for advanced students. But it can be recommend also to researchers in applied probability from economics and from natural sciences. The reader is only supposed to have a basic knowledge in probability theory. Nevertheless, some experience with stochastic processes is helpful.

The contents is divided into ten chapters: The first two chapters present basics: examples, connection between Lévy processes and infinitely divisible distributions. Chapter 3 treats the stable processes together with selfsimilar and selfdecomposable distributions. In Chapter 4 one finds the Lévy-Itô decomposition of sample functions of Lévy processes. Chapter 5 is devoted to properties of Lévy processes based on their distributions like moments, supports of the Lévy measures, continuity of the distributions. Subordination and time transformation is the topic of Chapter 6. In Chapter 7 the reader finds results on recurrence and transience of the underlying processes, and Chapter 8 contains the potential theory of Lévy processes, which is a generalization of the classical potential theory of the Laplacian, which is connected with the Brownian motion. Chapter 9 presents the Wiener-Hopf factorization and its consequences to short and long time behaviour of Lévy processes. The last Chapter 10 yields additional properties of infinitely divisible distributions.

Every chapter is finished by a series of ambitious exercises and notes, which include additional material, the solutions are given at the end of the book. The notes hint to further results in the literature and give insight into the history of the topics. More than five hundred references, all mentioned in the text, complete the survey on Lévy processes. This book should not be missed in any library having a department on probability theory and stochastic processes.

The contents is divided into ten chapters: The first two chapters present basics: examples, connection between Lévy processes and infinitely divisible distributions. Chapter 3 treats the stable processes together with selfsimilar and selfdecomposable distributions. In Chapter 4 one finds the Lévy-Itô decomposition of sample functions of Lévy processes. Chapter 5 is devoted to properties of Lévy processes based on their distributions like moments, supports of the Lévy measures, continuity of the distributions. Subordination and time transformation is the topic of Chapter 6. In Chapter 7 the reader finds results on recurrence and transience of the underlying processes, and Chapter 8 contains the potential theory of Lévy processes, which is a generalization of the classical potential theory of the Laplacian, which is connected with the Brownian motion. Chapter 9 presents the Wiener-Hopf factorization and its consequences to short and long time behaviour of Lévy processes. The last Chapter 10 yields additional properties of infinitely divisible distributions.

Every chapter is finished by a series of ambitious exercises and notes, which include additional material, the solutions are given at the end of the book. The notes hint to further results in the literature and give insight into the history of the topics. More than five hundred references, all mentioned in the text, complete the survey on Lévy processes. This book should not be missed in any library having a department on probability theory and stochastic processes.

Reviewer: Uwe Küchler (Berlin)