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Look Inside Stable Lévy Processes via Lamperti-Type Representations

Stable Lévy Processes via Lamperti-Type Representations

Part of Institute of Mathematical Statistics Monographs

Authors:
Andreas E. Kyprianou, University of Bath
Juan Carlos Pardo, Centro de Investigacion en Matematicas, A.C.

Date Published: June 2022
availability: Available
format: Hardback
isbn: 9781108480291

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About the Authors

Stable Lévy processes lie at the intersection of Lévy processes and self-similar Markov processes. Processes in the latter class enjoy a Lamperti-type representation as the space-time path transformation of so-called Markov additive processes (MAPs). This completely new mathematical treatment takes advantage of the fact that the underlying MAP for stable processes can be explicitly described in one dimension and semi-explicitly described in higher dimensions, and uses this approach to catalogue a large number of explicit results describing the path fluctuations of stable Lévy processes in one and higher dimensions. Written for graduate students and researchers in the field, this book systemically establishes many classical results as well as presenting many recent results appearing in the last decade, including previously unpublished material. Topics explored include first hitting laws for a variety of sets, path conditionings, law-preserving path transformations, the distribution of extremal points, growth envelopes and winding behaviour.
- This self-contained reference catalogues a wide range of results from the literature, including worked computations and new proofs
- Documents the last 15 years of development, presented by authors who helped clarify the theory through their research
- Written in a friendly, accessible style
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'This treatise takes readers on a superb journey through the fascinating worlds of stable Lévy processes and of a rich variety of further naturally related random processes. Andreas Kyprianou and Juan Carlos Pardo masterfully deploy an arsenal of techniques, which are already interesting on their own right, to reveal many classical or more recent high level results on the distributions of functionals and on the path behaviours stable processes. It is indeed remarkable that their methods lead to so many explicit formulas, some amazingly simple, some more complex. The authors should be praised for making accessible as a coherent whole a vast literature that has been developed over several decades, including the latest developments.' Jean Bertoin, University of Zurich

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Product details
- Date Published: June 2022
- format: Hardback
- isbn: 9781108480291
- length: 484 pages
- dimensions: 234 x 156 x 27 mm
- weight: 0.87kg
- availability: Available
Table of Contents
1. Stable distributions
2. Lévy processes
3. Stable processes
4. Hypergeometric Lévy processes
5. Positive self-similar Markov processes
6. Spatial fluctuations in one dimension
7. Doney–Kuznetsov factorisation and the maximum
8. Asymptotic behaviour for stable processes
9. Envelopes of positive self-similar Markov processes
10. Asymptotic behaviour for path transformations
11. Markov additive and self-similar Markov processes
12. Stable processes as self-similar Markov processes
13. Radial reflection and the deep factorisation
14. Spatial fluctuations and the unit sphere
15. Applications of radial excursion theory
16. Windings and up-crossings of stable processes
Appendix.
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Authors
Andreas E. Kyprianou, University of Bath
Andreas E. Kyprianou was educated at the University of Oxford and University of Sheffield and is currently a professor of mathematics at the University of Bath. He has spent over 25 years working on the theory and application of path-discontinuous stochastic processes and has over 130 publications, including a celebrated graduate textbook on Lévy processes. During his time in Bath, he co-founded and directed the Prob-L@B (Probability Laboratory at Bath), was PI for a multi-million-pound EPSRC Centre for Doctoral Training, and is currently the Director of the Bath Institute for Mathematical Innovation.
Juan Carlos Pardo, Centro de Investigacion en Matematicas, A.C.
Juan Carlos Pardo is a full professor at the department of Probability and Statistics at Centro de Investigación en Matemáticas (CIMAT). He was educated at the Universidad Nacional Autónoma de México (UNAM) and Université de Paris VI (Sorbonne Université). He has spent over 13 years working on the theory and application of path-discontinuous stochastic processes and has more than 50 publications in these areas. During the academic year 2018-2019, he held the David Parkin visiting professorship at the University of Bath.