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August 2016 Passage time and fluctuation calculations for subexponential Lévy processes
Ron Doney, Claudia Klüppelberg, Ross Maller
Bernoulli 22(3): 1491-1519 (August 2016). DOI: 10.3150/15-BEJ700

Abstract

We consider the passage time problem for Lévy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth on the positive side. Local and functional versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level. A significant connection is made with extreme value theory via regular variation or maximum domain of attraction conditions imposed on the positive tail of the canonical measure, which are shown to be necessary for the kind of convergence behaviour we are interested in.

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Ron Doney. Claudia Klüppelberg. Ross Maller. "Passage time and fluctuation calculations for subexponential Lévy processes." Bernoulli 22 (3) 1491 - 1519, August 2016. https://doi.org/10.3150/15-BEJ700

Information

Received: 1 April 2014; Revised: 1 October 2014; Published: August 2016
First available in Project Euclid: 16 March 2016

zbMATH: 1338.60127
MathSciNet: MR3474823
Digital Object Identifier: 10.3150/15-BEJ700

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

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Vol.22 • No. 3 • August 2016
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