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Exit Problems for Spectrally Negative Lévy Processes Reflected at Either the Supremum or the Infimum

Part of: Stochastic processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Xiaowen Zhou
Xiaowen Zhou*
Affiliation:
Concordia University
*
∗Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal, Quebec, H3G 1M8, Canada. Email address: [email protected]
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Abstract

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For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := S − X and Y := X − I first exit level a, respectively; let τ−(a) and κ−(a) denote the times when X first reaches Sτ(a) and Iκ(a), respectively. The main results of this paper concern the distributions of (τ(a), Sτ(a), τ−(a), Ŷτ(a)) and of (κ(a), Iκ(a), κ−(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.

Keywords

Spectrally negative Lévy processreflected Lévy processfluctuation theoryexit problemexcursionrisk modelruin time

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 1012 - 1030
Copyright
Copyright © Applied Probability Trust 2007 

Footnotes

Supported by an NSERC operating grant.

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