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Stability of the exit time for Lévy processes

Part of: Mathematical economics Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

Philip S. Griffin  and
Ross A. Maller
Philip S. Griffin*
Affiliation:
Syracuse University
Ross A. Maller*
Affiliation:
Australian National University
*
∗ Postal address: Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, USA. Email address: [email protected]
∗∗ Postal address: Centre for Financial Mathematics, and School of Finance, Actuarial Studies and Applied Statistics, Australian National University, Canberra, ACT 0200, Australia.
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Abstract

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This paper is concerned with the behaviour of a Lévy process when it crosses over a positive level, u, starting from 0, both as u becomes large and as u becomes small. Our main focus is on the time, τu, it takes the process to transit above the level, and in particular, on the stability of this passage time; thus, essentially, whether or not τu behaves linearly as u ↓ 0 or u → ∞. We also consider the conditional stability of τu when the process drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cramér condition.

Keywords

Lévy processpassage time above a levelstabilityinsurance risk processCramér conditionovershoot

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60K05: Renewal theory
Secondary: 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 712 - 734
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Research partially supported by ARC grant DP1092502.

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