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Wiener-Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Alan L. Lewis  and
Ernesto Mordecki
Alan L. Lewis*
Affiliation:
optioncity.net
Ernesto Mordecki*
Affiliation:
Universidad de la República
*
∗Postal address: 983 Bayside Cove, Newport Beach, CA 92660, USA. Email address: alewis@optioncity.net
∗∗Postal address: Facultad de Ciencias, Centro de Matemática, Universidad de la República, Iguá 4225, CP 11400 Montevideo, Uruguay. Email address: mordecki@cmat.edu.uy
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Abstract

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We show that the positive Wiener-Hopf factor of a Lévy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Lévy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.

Keywords

Lévy processWiener-Hopf factorizationBaxter-Donsker formularuin probabilityrational Laplace transformladder process

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60J50: Boundary theory
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 118 - 134
Copyright
Copyright © Applied Probability Trust 2008 

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