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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: e.j.baurdoux@lse.ac.uk
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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 

References

[1] Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.CrossRefGoogle Scholar
[2] Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
[3] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.CrossRefGoogle Scholar
[4] Chan, T. (2004). Some applications of Lévy processes in insurance and finance. Finance 25, 71–94.Google Scholar
[5] Chiu, S. N. and Yin, C. (2005). Passage times for a spectrally negative Lévy process with applications to risk theory. Bernoulli 11, 511–522.Google Scholar
[6] Doney, R. A. (1991). Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44, 566–576.CrossRefGoogle Scholar
[7] Emery, D. J. (1973). Exit problems for a spectrally positive process. Adv. Appl. Prob. 5, 498–520.CrossRefGoogle Scholar
[8] Gerber, H. U. (1990). When does the surplus reach a given target? Insurance Math. Econom. 9, 115–119.CrossRefGoogle Scholar
[9] Hubalek, F. and Kyprianou, A. E. (2007). Old and new examples of scale functions for spectrally negative Lévy processes. Preprint. Available at http://arxiv.org/abs/0801.0393.Google Scholar
[10] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397.CrossRefGoogle Scholar
[11] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities for competing claim processes. J. Appl. Prob. 41, 679–690.CrossRefGoogle Scholar
[12] 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
[13] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[14] Lévy, P. (1965). Processus Stochastiques et Mouvement Brownien, 2nd edn. Gauthier-Villars, Paris.Google Scholar
[15] Lundberg, F. (1903). Approximerad Framställning av Sannolikehets-Funktionen. Återförsäkering av Kollektivrisker. Almqvist & Wiksell, Uppsala.Google Scholar
[16] Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183–220.CrossRefGoogle Scholar
[17] Rogers, L. C. G. (2000). Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Prob. 37, 1173–1180.CrossRefGoogle Scholar
[18] Surya, B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135–149.CrossRefGoogle Scholar