Meromorphic levy processes and their fluctuation identities

A Kuznetsov, Andreas E Kyprianou, Juan-Carlos 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 Levy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Levy 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 Levy 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 Levy processes, which we call Meromorphic Levy 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 theirWiener.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.
Original languageEnglish
Pages (from-to)1101-1135
Number of pages35
JournalAnnals of Applied Probability
Volume22
Issue number3
DOIs
Publication statusPublished - Jun 2012

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Meromorphic
Lévy Process
Fluctuations
Factorization
Applied Probability
Risk Theory
Queuing Theory
Mathematical Finance
Infinite Interval
Stable Process
Stochastic Analysis
Laplace
Finance
Bernoulli
Insurance
Rational function
Random Field
Pole
Lévy process
Stochastic Processes

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Meromorphic levy processes and their fluctuation identities. / Kuznetsov, A; Kyprianou, Andreas E; Pardo, Juan-Carlos.

In: Annals of Applied Probability, Vol. 22, No. 3, 06.2012, p. 1101-1135.

Research output: Contribution to journalArticle

Kuznetsov, A ; Kyprianou, Andreas E ; Pardo, Juan-Carlos. / Meromorphic levy processes and their fluctuation identities. In: Annals of Applied Probability. 2012 ; Vol. 22, No. 3. pp. 1101-1135.
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