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The distribution and asympotic behaviour of the negative Wiener–Hopf factor for Lévy processes with rational positive jumps

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  11 December 2019

Ekaterina T. Kolkovska  and
Ehyter M. Martín-González
Ekaterina T. Kolkovska*
Affiliation:
Centro de Investigación en Matemáticas
Ehyter M. Martín-González*
Affiliation:
Universidad de Guanajuato
*
*Postal address: Área de Probabilidad y Estadística, Centro de Investigación en Matemáticas, A. P. 402, Guanajuato 36000, Mexico. Email address: todorova@cimat.mx
**Postal address: Departamento de Matemáticas, Universidad de Guanajuato, Mineral de Valenciana, Guanajuato 36240, Mexico.
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Abstract

We study the distribution of the negative Wiener–Hopf factor for a class of two-sided jump Lévy processes whose positive jumps have a rational Laplace transform. The positive Wiener–Hopf factor for this class of processes was studied by Lewis and Mordecki (2008). Here we obtain a formula for the Laplace transform of the negative Wiener–Hopf factor, as well as an explicit expression for its probability density in terms of sums of convolutions of known functions. Under additional regularity conditions on the Lévy measure of the studied processes, we also provide asymptotic results as $u\to-\infty$ for the distribution function F(u) of the negative Wiener–Hopf factor. We illustrate our results in some particular examples.

Keywords

Two-sided jump Lévy processWiener–Hopf factorizationnegative Wiener–Hopf factorLévy risk process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1086 - 1105
Copyright
© Applied Probability Trust 2019 

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