Stable Lévy Processes via Lamperti-Type Representations

Andreas Kyprianou and Juan Carlos Pardo offer an exceptional exploration of the captivating realms of stable Lévy processes. The book provides a comprehensive overview of stable processes and their connection with self-similar Markov processes as revealed by the Lamperti transform, unveiling advanced findings concerning functional distributions and fluctuation identities. As noted by the authors, the treatise should serve as material for a graduate course and as a standard reference for researchers working on stable Lévy processes. The latter are pervasive in a wide variety of scientific areas, including physics, biology, and economics, to name but a few. For a future edition, it would be great considering the addition of a motivational chapter to impress the reader with the effectiveness and elegance of such processes in describing natural phenomena.

The book has 16 chapters plus an Appendix reviewing more technical facts about complex analysis, special functions and the theory of Markov processes needed in the main text. I will briefly summarize each chapter as follows.

Chapters 1–3 review the fundamental concepts of the monograph: distributional stability, Lévy processes, the Wiener-Hopf factorization, and stable processes. Lévy (1925) introduced the class of stable distributions as limits of normalized sums of independent identically distributed random variables that do not satisfy a finite second-moment condition. The concept was fully developed a few years later (1937) by Lévy and Khintchine, independently. Since then, its impact in the scientific area has been noticeable. Lévy processes are the natural continuous-time analog of random walks, having stationary and independent increments. The Wiener-Hopf factorization provides distributional information about the local maxima and minima of the associated trajectories. Finally, stable processes naturally arise as scaling limits of random walks, a feature they share with the Brownian motion.

Chapters 4 and 5 are crucial for the entire monograph. Chapter 4 introduces the concept of a hypergeometric Lévy process. The key idea behind the latter is the specification of the Wiener-Hopf factors, appropriately chosen from the class of beta-subordinators. For the hypergeometric Lévy process, one can derive the explicit form of the first passage problem, the Mellin transform of exponential functionals, corresponding probability density functions, and, with fewer assumptions, distributional tails. Chapter 5 deals with the mathematical tool of positive self-similar Markov processes and their pathwise characterization as space-time changed Lévy processes through the Lamperti transform. The idea is that choosing an appropriate space-time transformation, self-similarity, and the property of having stationary and independent increments are interchangeable concepts.

Chapters 6 and 7 develop a suitable understanding of the distribution of a stable process and its extrema. Specifically, the explicit knowledge of the Lamperti transform leads to a set of fluctuation identities concerning, for instance, the first exit from an interval, the probability of hitting individual points in the interval before exiting it, or the first entrance into a bounded interval. Chapter 7 focuses on the law of the running supremum of a stable process. These two chapters are ancillary to Chapters 8–10, which develop integral tests determining the lower and upper envelopes of the sample paths and path functionals of stable processes when time goes to zero (or infinity).

Starting from Chapter 11, the authors extend the analysis to self-similar Markov processes exploring the real line, that is, not bounded to remain positive. It turns out that such processes tightly connect to a class of processes with Lévy dynamics whose characteristics change each time an independent Markov chain changes state. Such processes are named the Markov additive processes (MAPs). The main finding is that one can express a self-similar Markov process as a MAP via the Lamperti-Kiu space-time transform. Roughly speaking, the Lamperti-Kiu transform plays for a real self-similar Markov process the same role the Lamperti transform plays for the positive case. The highlight of Chapter 12 is the Riesz–Bogdan–$\stackrel{.}{\text{Z}}$ ak transform of a one-dimensional stable process with two-sided jumps (Theorem 12.7) and of a d-dimensional isotropic stable process with d larger than one (Theorem 12.8). The transform turns out to be an essential tool for deriving fluctuation identities. Chapter 13 delves into the so-called deep factorization of a one-dimensional stable process with positive and negative jumps. Remarkably, the introduction of deep factorization provided the first explicit example of the Wiener-Hopf factorization of a MAP.

Chapters 14 and 15 provide explicit fluctuation identities for the isotropic stable processes in higher dimensions. Chapter 16 concludes by exploring the extension of Spitzer’s 1958 result on the winding number, that is, the net position of winding clockwise and unwinding anti-clockwise around the origin of a planar Brownian motion to the case of isotropic planar stable processes.

This book is a beautiful journey in a field that reveals the intimate relationships between self-similarity and stable processes, as Lamperti first showed in his pathbreaking paper (Lamperti 1972). One may be fascinated by the variety of techniques employed by the authors or by the impressive explicit identities worked out in the text. With higher probability, what will impress the reader the most is the beauty of self-similarity and scaling behavior that the book deeply unravels and leads back to the invariance features of stable Lévy processes. This textbook will become a standard reference in the field.

Giacomo Bormetti
Department of Mathematics
Alma Mater Studiorum University of Bologna
Bologna, Italy
[email protected]