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On the occupation times in a delayed Sparre Andersen risk model with exponential claims. (English) Zbl 1371.91094

Summary: In this paper, we study the joint Laplace transform of the occupation times in disjoint intervals until ruin in a delayed Sparre Andersen risk model with general inter-claim times and exponential claims. We extend the transformation method in the literature and apply the theoretical fluctuation techniques to derive an explicit expression of the joint Laplace transform under consideration. Further, with the presence of a constant dividend barrier, we derive explicit expressions for the Laplace transforms of the time of ruin and the non-dividend paying duration, namely the total length of non-dividend paying periods prior to ruin. This quantity is of practical interest but has not been studied in the literature to date. Within this paper, all of the Laplace transforms are expressed in terms of scale functions associated with the given spectrally negative Lévy process. Numerical examples are also provided at the end of this paper regarding the Laplace transform of the non-dividend paying duration to illustrate how the distribution of this occupation time behaves in response to varying parameters and the impact of delay on the occupation times comparing with an ordinary Sparre Andersen risk model.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G51 Processes with independent increments; Lévy processes
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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[1] Albrecher, H.; Claramunt, M. M.; Mármol, M., On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang (n) interclaim times, Insurance Math. Econom., 37, 2, 324-334 (2005) · Zbl 1117.91377
[2] Albrecher, H.; Hartinger, J., A risk model with multilayer dividend strategy, N. Am. Actuar. J., 11, 2, 43-64 (2007) · Zbl 1480.91178
[3] Albrecher, H.; Hartinger, J.; Tichy, R. F., On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier, Scand. Actuar. J., 2005b, 2, 103-126 (2005) · Zbl 1092.91036
[4] Albrecher, H.; Ivanovs, J., A risk model with an observer in a Markov environment, Risks, 1, 3, 148-161 (2013)
[5] Albrecher, H.; Kainhofer, R., Risk theory with a nonlinear dividend barrier, Computing, 68, 4, 289-311 (2002) · Zbl 1076.91521
[6] Albrecher, H.; Kainhofer, R.; Tichy, R. F., Simulation methods in ruin models with non-linear dividend barriers, Math. Comput. Simulation, 62, 3, 277-287 (2003) · Zbl 1036.91029
[7] Badescu, A.; Drekic, S.; Landriault, D., Analysis of a threshold dividend strategy for a MAP risk model, Scand. Actuar. J., 2007, 4, 227-247 (2007) · Zbl 1164.91024
[8] Borovkov, K. A.; Dickson, D. C., On the ruin time distribution for a Sparre Andersen process with exponential claim sizes, Insurance Math. Econom., 42, 3, 1104-1108 (2008) · Zbl 1141.91486
[9] Breuer, L., Occupation times for Markov-modulated Brownian motion, J. Appl. Probab., 49, 2, 549-565 (2012) · Zbl 1258.60046
[11] Dickson, D. C.; Egídio Dos Reis, A. D., On the distribution of the duration of negative surplus, Scand. Actuar. J., 1996, 2, 148-164 (1996) · Zbl 0864.62069
[12] Egídio Dos Reis, A., How long is the surplus below zero?, Insurance Math. Econom., 12, 1, 23-38 (1993) · Zbl 0777.62096
[13] Gerber, H. U., On the probability of ruin in the presence of a linear dividend barrier, Scand. Actuar. J., 1981, 2, 105-115 (1981) · Zbl 0455.62086
[14] Gerber, H. U., When does the surplus reach a given target?, Insurance Math. Econom., 9, 2, 115-119 (1990) · Zbl 0731.62153
[15] Gerber, H. U.; Shiu, E. S., The time value of ruin in a Sparre Andersen model, N. Am. Actuar. J., 9, 2, 49-69 (2005) · Zbl 1085.62508
[16] Huang, Y.; Yu, W., The Gerber-Shiu discounted penalty function of Sparre Andersen risk model with a constant dividend barrier, Math. Probl. Eng. (2014) · Zbl 1407.91139
[17] Kuznetsov, A.; Kyprianou, A. E.; Rivero, V., The theory of scale functions for spectrally negative Lévy processes, (Lévy Matters II (2013), Springer), 97-186 · Zbl 1261.60047
[18] Kyprianou, A. E., Introductory Lectures on Fluctuations of Lévy Processes with Applications (2006), Springer Science & Business Media · Zbl 1104.60001
[19] Kyprianou, A. E., Gerber-Shiu Risk Theory (2013), Springer Science & Business Media · Zbl 1277.91003
[20] Landriault, D., Constant dividend barrier in a risk model with interclaim-dependent claim sizes, Insurance Math. Econom., 42, 1, 31-38 (2008) · Zbl 1141.91523
[21] 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
[22] Landriault, D.; Shi, T., Occupation times in the MAP risk model, Insurance Math. Econom., 60, 75-82 (2015) · Zbl 1308.91087
[23] Li, S.; Garrido, J., On a class of renewal risk models with a constant dividend barrier, Insurance Math. Econom., 35, 3, 691-701 (2004) · Zbl 1122.91345
[24] Li, S.; Garrido, J., On ruin for the Erlang (n) risk process, Insurance Math. Econom., 34, 3, 391-408 (2004) · Zbl 1188.91089
[25] Li, S.; Lu, Y., The distribution of total dividend payments in a Sparre Andersen model, Statist. Probab. Lett., 79, 9, 1246-1251 (2009) · Zbl 1160.62359
[26] Li, Y.; Zhou, X., On pre-exit joint occupation times for spectrally negative Lévy processes, Statist. Probab. Lett., 94, 48-55 (2014) · Zbl 1315.60049
[27] Li, B.; Zhou, X., The joint Laplace transforms for diffusion occupation times, Adv. Appl. Probab., 45, 4, 1049-1067 (2013) · Zbl 1370.60136
[28] Lin, X. S.; Pavlova, K. P., The compound Poisson risk model with a threshold dividend strategy, Insurance Math. Econom., 38, 1, 57-80 (2006) · Zbl 1157.91383
[29] Lin, X. S.; Willmot, G. E.; Drekic, S., The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function, Insurance Math. Econom., 33, 3, 551-566 (2003) · Zbl 1103.91369
[30] 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
[31] Ramaswami, V., Passage times in fluid models with application to risk processes, Methodol. Comput. Appl. Probab., 8, 4, 497-515 (2006) · Zbl 1110.60067
[32] Shi, T.; Landriault, D., Distribution of the time to ruin in some Sparre Andersen risk models, ASTIN Bull., 43, 01, 39-59 (2013) · Zbl 1284.91270
[33] Yuen, K. C.; Wang, G.; Li, W. K., The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier, Insurance Math. Econom., 40, 1, 104-112 (2007) · Zbl 1273.91456
[34] Zhao, C.; Zhang, C., Joint density of the number of claims until ruin and the time to ruin in the delayed renewal risk model with Erlang (n) claims, J. Comput. Appl. Math., 244, 102-114 (2013) · Zbl 1264.91076
[35] Zhou, X., On a classical risk model with a constant dividend barrier, N. Am. Actuar. J., 9, 4, 95-108 (2005) · Zbl 1215.60051
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