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Last exit before an exponential time for spectrally negative Lévy processes. (English) Zbl 1170.60020

Summary: S. N. Chiu and C. Yin [Bernoulli 11, No. 3, 511–522 (2005; Zbl 1076.60038)] found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to \(\infty\), 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 \(\infty \), 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 [R. A. Doney, J. Lond. Math. Soc., II. Ser. 44, No. 3, 566–576 (1991; Zbl 0699.60061)].

MSC:

60G51 Processes with independent increments; Lévy processes
91B30 Risk theory, insurance (MSC2010)
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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. · Zbl 1075.60037
[2] Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121 ). Cambridge University Press. · Zbl 0861.60003
[3] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766. JSTOR: · Zbl 0322.60068
[4] Chan, T. (2004). Some applications of Lévy processes in insurance and finance. Finance 25, 71–94.
[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. · Zbl 1076.60038
[6] Doney, R. A. (1991). Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44, 566–576. · Zbl 0699.60061
[7] Emery, D. J. (1973). Exit problems for a spectrally positive process. Adv. Appl. Prob. 5, 498–520. JSTOR: · Zbl 0297.60035
[8] Gerber, H. U. (1990). When does the surplus reach a given target? Insurance Math. Econom. 9, 115–119. · Zbl 0731.62153
[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. · Zbl 1274.60148
[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. · Zbl 1061.60075
[11] Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities for competing claim processes. J. Appl. Prob. 41, 679–690. · Zbl 1065.60100
[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. · Zbl 1066.60049
[13] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin. · Zbl 1104.60001
[14] Lévy, P. (1965). Processus Stochastiques et Mouvement Brownien , 2nd edn. Gauthier-Villars, Paris. · Zbl 0248.60004
[15] Lundberg, F. (1903). Approximerad Framställning av Sannolikehets-Funktionen. Återförsäkering av Kollektivrisker . Almqvist & Wiksell, Uppsala.
[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. · Zbl 1049.60042
[17] Rogers, L. C. G. (2000). Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Prob. 37, 1173–1180. · Zbl 0981.60048
[18] Surya, B. A. (2008). Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Prob. 45, 135–149. · Zbl 1140.60027
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