Last exit before an exponential time for spectrally negative Lévy processes
E. J. Baurdoux
Source: J. Appl. Probab. Volume 46, Number 2 (2009), 542-558.
Abstract
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).
Primary Subjects: 60G99
Secondary Subjects: 91B30
Keywords: Spectrally negative Lévy process; exit problem; fluctuation theory; last passage time; risk theory
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Permanent link to this document: http://projecteuclid.org/euclid.jap/1245676105
Digital Object Identifier: doi:10.1239/jap/1245676105
References
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.
Mathematical Reviews (MathSciNet):
MR2024845
Digital Object Identifier: doi:10.1016/j.spa.2003.07.005
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.
Mathematical Reviews (MathSciNet):
MR1406564
Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705--766.
Mathematical Reviews (MathSciNet):
MR386027
Digital Object Identifier: doi:10.2307/1426397
JSTOR: links.jstor.org
Zentralblatt MATH:
0322.60068
Chan, T. (2004). Some applications of Lévy processes in insurance and finance. Finance 25, 71--94.
Chiu, S. N. and Yin, C. (2005). Passage times for a spectrally negative Lévy process with applications to risk theory. Bernoulli 11, 511--522.
Mathematical Reviews (MathSciNet):
MR2146892
Digital Object Identifier: doi:10.3150/bj/1120591186
Project Euclid: euclid.bj/1120591186
Doney, R. A. (1991). Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44, 566--576.
Mathematical Reviews (MathSciNet):
MR1149016
Digital Object Identifier: doi:10.1112/jlms/s2-44.3.566
Emery, D. J. (1973). Exit problems for a spectrally positive process. Adv. Appl. Prob. 5, 498--520.
Mathematical Reviews (MathSciNet):
MR341623
Digital Object Identifier: doi:10.2307/1425831
JSTOR: links.jstor.org
Zentralblatt MATH:
0297.60035
Gerber, H. U. (1990). When does the surplus reach a given target? Insurance Math. Econom. 9, 115--119.
Mathematical Reviews (MathSciNet):
MR1084494
Zentralblatt MATH:
0731.62153
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.
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.
Mathematical Reviews (MathSciNet):
MR2071427
Digital Object Identifier: doi:10.1214/105051604000000332
Project Euclid: euclid.aoap/1089736289
Zentralblatt MATH:
1061.60075
Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities for competing claim processes. J. Appl. Prob. 41, 679--690.
Mathematical Reviews (MathSciNet):
MR2074816
Digital Object Identifier: doi:10.1239/jap/1091543418
Project Euclid: euclid.jap/1091543418
Zentralblatt MATH:
1065.60100
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.
Mathematical Reviews (MathSciNet):
MR2099651
Digital Object Identifier: doi:10.1214/105051604000000927
Project Euclid: euclid.aoap/1099674077
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR2250061
Lévy, P. (1965). Processus Stochastiques et Mouvement Brownien, 2nd edn. Gauthier-Villars, Paris.
Mathematical Reviews (MathSciNet):
MR190953
Lundberg, F. (1903). Approximerad Framställning av Sannolikehets-Funktionen. Återförsäkering av Kollektivrisker. Almqvist & Wiksell, Uppsala.
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.
Mathematical Reviews (MathSciNet):
MR2054585
Digital Object Identifier: doi:10.1023/B:JOTP.0000020481.14371.37
Rogers, L. C. G. (2000). Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Prob. 37, 1173--1180.
Mathematical Reviews (MathSciNet):
MR1808924
Digital Object Identifier: doi:10.1239/jap/1014843099
Project Euclid: euclid.jap/1014843099
Surya, B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135--149.
Mathematical Reviews (MathSciNet):
MR2409316
Digital Object Identifier: doi:10.1239/jap/1208358957
Project Euclid: euclid.jap/1208358957
Journal of Applied Probability
