The Annals of Applied Probability

Meromorphic Lévy processes and their fluctuation identities

A. Kuznetsov, A. E. Kyprianou, and J. C. Pardo

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Abstract

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis, Random Fields and Applications (2011) 119–146, Electron. J. Probab. 13 (2008) 1672–1701], hyper-exponential and generalized hyper-exponential Lévy processes [Quant. Finance 10 (2010) 629–644], Lamperti-stable processes in [J. Appl. Probab. 43 (2006) 967–983, Probab. Math. Statist. 30 (2010) 1–28, Stochastic Process. Appl. 119 (2009) 980–1000, Bull. Sci. Math. 133 (2009) 355–382], Hypergeometric processes in [Ann. Appl. Probab. 20 (2010) 522–564, Ann. Appl. Probab. 21 (2011) 2171–2190, Bernoulli 17 (2011) 34–59], β-processes in [Ann. Appl. Probab. 20 (2010) 1801–1830] and θ-processes in [J. Appl. Probab. 47 (2010) 1023–1033].

In this paper we introduce a new family of Lévy processes, which we call Meromorphic Lévy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of their Wiener–Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Laplace exponent, all of which are real numbers. The specific structure of the M-class Wiener–Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance, insurance risk theory and queuing theory.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 3 (2012), 1101-1135.

Dates
First available in Project Euclid: 18 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1337347540

Digital Object Identifier
doi:10.1214/11-AAP787

Mathematical Reviews number (MathSciNet)
MR2977987

Zentralblatt MATH identifier
1252.60044

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60G50: Sums of independent random variables; random walks

Keywords
Lévy processes Wiener–Hopf factorization exit problems fluctuation theory

Citation

Kuznetsov, A.; Kyprianou, A. E.; Pardo, J. C. Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22 (2012), no. 3, 1101--1135. doi:10.1214/11-AAP787. https://projecteuclid.org/euclid.aoap/1337347540


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