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Last Exit Before an Exponential Time for Spectrally Negative Lévy Processes

Part of: Stochastic processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

E. J. Baurdoux
E. J. Baurdoux*
Affiliation:
Universiteit Utrecht
*
∗Current address: Department of Statistics, London School of Economics, Houghton Street, London, WC2A 2AE, UK. Email address: [email protected]
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Abstract

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Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).

Keywords

Spectrally negative Lévy processexit problemfluctuation theorylast passage timerisk theory

MSC classification

Secondary: 60G99: None of the above, but in this section 91B30: Risk theory, insurance
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 542 - 558
Copyright
Copyright © Applied Probability Trust 2009 

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