Stable Lévy Processes via Lamperti-Type Representations
Stable Lévy Processes via Lamperti-Type Representations
Juan Carlos Pardo , Centro de Investigacion en Matematicas, A.C.
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USDStable 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many classical results as well as presenting many recent results appearing in the last decade, including previously unpublished material.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
Reviews & endorsements
'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
Product details
March 2022Adobe eBook Reader
9781108572163
0 pages
This ISBN is for an eBook version which is distributed on our behalf by a third party.
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.
