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Parisian ruin for a refracted Lévy process. (English) Zbl 1394.60046
Summary: In this paper, we investigate Parisian ruin for a Lévy surplus process with an adaptive premium rate, namely a refracted Lévy process. Our main contribution is a generalization of the result in [R. Loeffen et al., Bernoulli 19, No. 2, 599–609 (2013; Zbl 1267.60054)] for the probability of Parisian ruin of a standard Lévy insurance risk process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Despite the more general setup considered here, our main result is as compact and has a similar structure. Examples are provided.

MSC:
60G51 Processes with independent increments; Lévy processes
91B30 Risk theory, insurance (MSC2010)
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[1] Albrecher, H.; Ivanovs, J., Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations, Stochastic Process. Appl., 127, 2, 643-656, (2017) · Zbl 1354.60048
[2] Baurdoux, E. J.; Pardo, J. C.; Pérez, J. L.; Renaud, J.-F., Gerber-shiu distribution at Parisian ruin for Lévy insurance risk processes, J. Appl. Probab., 53, 2, 572-584, (2016) · Zbl 1344.60046
[3] Czarna, I.; Palmowski, Z., Ruin probability with Parisian delay for a spectrally negative Lévy risk process, J. Appl. Probab., 48, 4, 984-1002, (2011) · Zbl 1232.60036
[4] Egami, M.; Yamazaki, K., Phase-type Fitting of scale functions for spectrally negative Lévy processes, J. Comput. Appl. Math., 264, 1-22, (2014) · Zbl 1291.60094
[5] Feller, W., An introduction to probability theory and its applications. vol. II, (1971), John Wiley & Sons Inc. New York · Zbl 0219.60003
[6] Guérin, H.; Renaud, J.-F., On the distribution of cumulative Parisian ruin, Insurance Math. Econom., 73C, 116-123, (2017) · Zbl 1397.91285
[7] Kuznetsov, A.; Kyprianou, A. E.; Rivero, V., The theory of scale functions for spectrally negative Lévy processes, Lévy Matters - Springer Lecture Notes in Mathematics, (2012) · Zbl 1261.60047
[8] Kyprianou, A. E., Fluctuations of Lévy processes with applications - introductory lectures, (2014), Universitext, Springer Heidelberg · Zbl 1384.60003
[9] Kyprianou, A. E.; Loeffen, R. L., Refracted Lévy processes, Ann. Inst. Henri Poincaré Probab. Stat., 46, 1, 24-44, (2010) · Zbl 1201.60042
[10] Landriault, D.; Li, B.; Zhang, H., On magnitude, asymptotics and duration of drawdowns for Lévy models, Bernoulli, 23, 1, 432-458, (2017) · Zbl 1407.60067
[11] Landriault, D.; Renaud, J.-F.; Zhou, X., Occupation times of spectrally negative Lévy processes with applications, Stochastic Process. Appl., 121, 11, 2629-2641, (2011) · Zbl 1227.60061
[12] Landriault, D.; Renaud, J.-F.; Zhou, X., An insurance risk model with Parisian implementation delays, Methodol. Comput. Appl. Probab., 16, 3, 583-607, (2014) · Zbl 1319.60098
[13] Loeffen, R.L., 2015. On obtaining simple identities for overshoots of spectrally negative Lévy processes, ArXiv:1410.5341v2 [Math.PR]
[14] Loeffen, R. L.; Czarna, I.; Palmowski, Z., Parisian ruin probability for spectrally negative Lévy processes, Bernoulli, 19, 2, 599-609, (2013) · Zbl 1267.60054
[15] Loeffen, R. L.; Renaud, J.-F.; Zhou, X., Occupation times of intervals until first passage times for spectrally negative Lévy processes, Stochastic Process. Appl., 124, 3, 1408-1435, (2014) · Zbl 1287.60062
[16] Renaud, J.-F., On the time spent in the red by a refracted Lévy risk process, J. Appl. Probab., 51, 4, 1171-1188, (2014) · Zbl 1321.60099
[17] Wong, J. T.Y.; Cheung, E. C.K., On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps, Insurance Math. Econom., 65, 280-290, (2015) · Zbl 1348.91189
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