The Annals of Probability

On extrema of stable processes

Alexey Kuznetsov

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Abstract

We study the Wiener–Hopf factorization and the distribution of extrema for general stable processes. By connecting the Wiener–Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as infinite series representations and asymptotic expansions for the density of supremum, explicit expressions for the Wiener–Hopf factors and the Mellin transform of the supremum, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum functional.

Article information

Source
Ann. Probab., Volume 39, Number 3 (2011), 1027-1060.

Dates
First available in Project Euclid: 16 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1300281731

Digital Object Identifier
doi:10.1214/10-AOP577

Mathematical Reviews number (MathSciNet)
MR2789582

Zentralblatt MATH identifier
1218.60037

Subjects
Primary: 60G52: Stable processes

Keywords
Stable processes supremum Wiener–Hopf factorization Mellin transform functional equations elliptic functions double Gamma function q-Pochhammer symbol Clausen function

Citation

Kuznetsov, Alexey. On extrema of stable processes. Ann. Probab. 39 (2011), no. 3, 1027--1060. doi:10.1214/10-AOP577. https://projecteuclid.org/euclid.aop/1300281731


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Supplemental materials

  • Supplementary material: Appendix A: Detailed proofs of some results related to the double gamma function. This supplement material provides detailed computations needed to derive formulas (4.10), (4.11), (7.1), (7.2), (7.5) and to prove Corollary 3 and Theorem 8.