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Gerber-Shiu distribution at Parisian ruin for Lévy insurance risk processes. (English) Zbl 1344.60046

Summary: Inspired by the works of D. Landriault et al. [Stochastic Processes Appl. 121, No. 11, 2629–2641 (2011; Zbl 1227.60061); Methodol. Comput. Appl. Probab. 16, No. 3, 583–607 (2014; Zbl 1319.60098)], we study the Gerber-Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber-Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again, then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of so-called scale functions. In particular, we extend the recent results of Landriault et al. [loc. cit.].

MSC:

60G51 Processes with independent increments; Lévy processes
60J99 Markov processes
91B30 Risk theory, insurance (MSC2010)
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References:

[1] Albrecher, H., Ivanovs, J. and Zhou, X. (2016). Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 1364-1382. · Zbl 1338.60125
[2] Bertoin, J. (1996). Lévy Processes . Cambridge University Press. · Zbl 0861.60003
[3] Biffis, E. and Kyprianou, A. E. (2010). A note on scale functions and time value of ruin for Lévy insurance risk processes. Insurance Math. Econom. 46, 85-91. · Zbl 1231.91145
[4] Chesney, M., Jeanblanc-Picqué, M., and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29, 165-184. · Zbl 0882.60042
[5] Czarna, I. (2016). Parisian ruin probability with a lower ultimate bankrupt barrier. Scand. Actuarial J. 2016, 319-337. · Zbl 1401.91124
[6] Czarna, I. and Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk process. J. Appl. Prob. 48, 984-1002. · Zbl 1232.60036
[7] Dassios, A. and Wu, S. (2008). Parisian ruin with exponential claims. Unpublished manuscript.
[8] Gerber, H. U. and Shiu, E. S. W. (1997). The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance Math. Econom. 21, 129-137. · Zbl 0894.90047
[9] Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48-78. · Zbl 1081.60550
[10] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications , 2nd edn. Springer, Heidelberg.
[11] Landriault, D., Renaud, J.-F. and Zhou, X. (2011). Occupation times of spectrally negative Lévy processes with applications. Stoch. Process. Appl. 121, 2629-2641. · Zbl 1227.60061
[12] Landriault, D., Renaud, J.-F. and Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Methodol. Comput. Appl. Prob. 16, 583-607. · Zbl 1319.60098
[13] Loeffen, R., Czarna, I. and Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19, 599-609. · Zbl 1267.60054
[14] Loeffen, R. L., Renaud, J.-F. and Zhou, X. (2014). Occupation times of intervals until first passage times for spectrally negative Lévy processes. Stoch. Process. Appl. 124, 1408-1435. · Zbl 1287.60062
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