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Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

A. Kuznetsov
A. Kuznetsov*
Affiliation:
York University
*
∗Postal address: Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada. Email address: [email protected]
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Abstract

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In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.

Keywords

Lévy processWiener-Hopf factorizationmeromorphic functioninfinite producttheta function

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60E10: Characteristic functions; other transforms
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1023 - 1033
Copyright
Copyright © Applied Probability Trust 2010 

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