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Asymptotics for the First Passage Times of Lévy Processes and Random Walks

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  30 January 2018

Denis Denisov  and
Vsevolod Shneer
Denis Denisov*
Affiliation:
Cardiff University
Vsevolod Shneer*
Affiliation:
Heriot-Watt University
*
∗ Postal address: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK.
∗∗ Postal address: Department of AMS, Heriot-Watt University, Edinburgh, EH14 4AS, UK. Email address: v.shneer@hw.ac.uk
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Abstract

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We study the exact asymptotics for the distribution of the first time, τx , a Lévy process X t crosses a fixed negative level -x. We prove that ℙ{τx >t} ~V(x) ℙ{X t ≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx >t} explicitly in both light- and heavy-tailed cases.

Keywords

Lévy processrandom walkbusy periodfirst passage timesubexponential distributionlarge deviationsingle-server queue

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; L'evy processes
Secondary: 60K25: Queueing theory

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 64 - 84
Copyright
Copyright © Applied Probability Trust 2013 

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