The Annals of Applied Probability

Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes

Alexey Kuznetsov

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Abstract

In this paper we introduce a ten-parameter family of Lévy processes for which we obtain Wiener–Hopf factors and distribution of the supremum process in semi-explicit form. This family allows an arbitrary behavior of small jumps and includes processes similar to the generalized tempered stable, KoBoL and CGMY processes. Analytically it is characterized by the property that the characteristic exponent is a meromorphic function, expressed in terms of beta and digamma functions. We prove that the Wiener–Hopf factors can be expressed as infinite products over roots of a certain transcendental equation, and the density of the supremum process can be computed as an exponentially converging infinite series. In several special cases when the roots can be found analytically, we are able to identify the Wiener–Hopf factors and distribution of the supremum in closed form. In the general case we prove that all the roots are real and simple, and we provide localization results and asymptotic formulas which allow an efficient numerical evaluation. We also derive a convergence acceleration algorithm for infinite products and a simple and efficient procedure to compute the Wiener–Hopf factors for complex values of parameters. As a numerical example we discuss computation of the density of the supremum process.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 5 (2010), 1801-1830.

Dates
First available in Project Euclid: 25 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1282747401

Digital Object Identifier
doi:10.1214/09-AAP673

Mathematical Reviews number (MathSciNet)
MR2724421

Zentralblatt MATH identifier
1222.60038

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60E10: Characteristic functions; other transforms

Keywords
Lévy process supremum process Wiener–Hopf factorization meromorphic function infinite product

Citation

Kuznetsov, Alexey. Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20 (2010), no. 5, 1801--1830. doi:10.1214/09-AAP673. https://projecteuclid.org/euclid.aoap/1282747401


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