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Stability of the exit time for Lévy processes

Part of: Stochastic processes Mathematical economics 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: psgriffi@syr.edu
∗∗ 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.

References

Asmussen, S. (1982). Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue. Adv. Appl. Prob. 14, 143–170.Google Scholar
Asmussen, S. (1984). Approximations for the probability of ruin within finite time. Scand. Actuarial J. 1984, 31–57. (Correction: 1985 (1985), 64.)CrossRefGoogle Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64, 103–125.CrossRefGoogle Scholar
Avram, F. and Usabel, M. (2003). Finite time ruin probabilities with one Laplace inversion. Insurance Math. Econom. 32, 371–377.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. (1997). Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121, 345–354.Google Scholar
Bertoin, J. and Doney, R. A. (1994). Cramér's estimate for Lévy processes. Statist. Prob. Lett. 21, 363–365.Google Scholar
Bertoin, J., van Harn, K. and Steutel, F. W. (1999). Renewal theory and level passage by subordinators. Statist. Prob. Lett. 45, 65–69.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Braverman, M. (1997). Suprema and sojourn times of Lévy processes with exponential tails. Stoch. Proc. Appl. 68, 265–283.Google Scholar
Doney, R. A. (2004). Stochastic bounds for Lévy Processes. Ann. Prob. 32, 1545–1552.CrossRefGoogle Scholar
Doney, R. A. (2005). Fluctuation Theory for Lévy Processes (Lecture Notes Math. 1897). Springer, Berlin.Google Scholar
Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, 91–106.Google Scholar
Doney, R. A. and Maller, R. A. (2002). Stability and attraction to normality for Lévy processes at zero and infinity. J. Theoret. Prob. 15, 751–792.Google Scholar
Doney, R. A. and Maller, R. A. (2002). Stability of the overshoot for Lévy processes. Ann. Prob. 30, 188–212.Google Scholar
Doney, R. A. and Maller, R. A. (2004). Moments of passage times for Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 40, 279–297.Google Scholar
Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13, 37–53.CrossRefGoogle Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.CrossRefGoogle Scholar
Griffin, P. S. and Maller, R. A. (2011). Path decomposition of ruinous behaviour for a general Lévy insurance risk process. To appear in Ann. Appl. Prob. Google Scholar
Griffin, P. S. and Maller, R. A. (2011). The time at which a Lévy process creeps. Submitted.Google Scholar
Gut, A. (2009). Stopped Random Walks, 2nd edn. Springer, New York.Google Scholar
Hall, W. J. (1970). On Wald's equations in continuous time. J. Appl. Prob. 7, 59–68.Google Scholar
Hao, X. and Tang, Q. (2009). Asymptotic ruin probabilities of the Lévy insurance model under periodic taxation. ASTIN Bull. 39, 479–494.CrossRefGoogle Scholar
Heyde, C. C. and Wang, D. (2009). Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims. Adv. Appl. Prob., 41, 206–224.CrossRefGoogle Scholar
Kesten, H. and Maller, R. (1999). Stability and other limit laws for exit times of random walks from a strip or a halfplane. Ann. Inst. H. Poincaré Prob. Statist. 35, 685–734.CrossRefGoogle Scholar
Klüppelberg, C., Kyprianou, A.E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.CrossRefGoogle Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Lai, T. L. (1975). On uniform integrability in renewal theory. Bull. Inst. Math. Acad. Sinica 3, 99–105.Google Scholar
Park, H. S. and Maller, R. A. (2008). Moment and MGF convergence of overshoots and undershoots for Lévy insurance risk processes. Adv. Appl. Prob. 40, 716–733.Google Scholar
Percheskii, E. A. and Rogozin, B. A. (1969). On Joint distributions of random variables associated with fluctuations of a process with independent increments, Theoret. Prob. Appl. 14, 410–423.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar