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On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula. (English) Zbl 1151.91565

Summary: We consider an extension to the classical compound Poisson risk model in which we introduce a dependence structure between the claim amounts and the interclaim time. This structure is embedded via a generalized Farlie-Gumbel-Morgenstern copula. In this framework, we derive the Laplace transform of the Gerber-Shiu discounted penalty function. An explicit expression for the Laplace transform of the time of ruin is given for exponential claim sizes.

MSC:

91B30 Risk theory, insurance (MSC2010)
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[1] Albrecher, H.; Teugels, J., Exponential behavior in the presence of dependence in risk theory, Journal of Applied Probability, 43, 1, 265-285 (2006)
[2] Boudreault, M.; Cossette, H.; Landriault, D.; Marceau, E., On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian Actuarial Journal, 301-323 (2006) · Zbl 1145.91030
[3] Bouye, E., Nikeghbali, A., Riboulet, G., Roncalli, T., 2000. Copulas for Finance. A reading guide and some applications. Rapport technique du Groupe de recherche opérationnelle, Crédit Lyonnais; Bouye, E., Nikeghbali, A., Riboulet, G., Roncalli, T., 2000. Copulas for Finance. A reading guide and some applications. Rapport technique du Groupe de recherche opérationnelle, Crédit Lyonnais
[4] Denuit, M.; Dhaene, J.; Goovaerts, M. J.; Kaas, R., Actuarial Theory for Dependent Risks—Measures, Orders and Models (2005), Wiley: Wiley New York
[5] Dickson, D. C.M.; Hipp, C., On the time to ruin for Erlang(2) risk process, Insurance: Mathematics and Economics, 29, 333-344 (2001) · Zbl 1074.91549
[6] Drouet Mari, D.; Kotz, S., Correlation and Dependence (2001), Imperial College Press · Zbl 0977.62004
[7] Frees, E. W.; Valdez, E. A., Understanding relationships using copulas, North American Actuarial Journal, 2, 1-25 (1998) · Zbl 1081.62564
[8] Gerber, H. U.; Shiu, E. S.W, On the time value of ruin, North American Actuarial Journal, 2, 48-78 (1998) · Zbl 1081.60550
[9] Joe, H., Multivariate Models and Dependence Concepts (1997), Chapman and Hall: Chapman and Hall London · Zbl 0990.62517
[10] Klimenok, V., On the modification of Rouche’s theorem for the queuing theory problems, Queuing Systems, 38, 431-434 (2001) · Zbl 1079.90523
[11] Li, S.; Garrido, J., On ruin for the Erlang(n) risk process, Insurance: Mathematics and Economics, 34, 391-408 (2004) · Zbl 1188.91089
[12] McNeil, A.; Frey, R.; Embretchs, P., Quantitative Risk Management (2005), Princeton Press: Princeton Press Princeton
[13] Nelsen, R. B., (An Introduction to Copulas. An Introduction to Copulas, Springer Series in Statistics. (2006), Springer-Verlag: Springer-Verlag New York) · Zbl 1152.62030
[14] Rodriguez-Lallena, J. A.; Ubena-Flores, M., A new class of bivariate copulas, Statistics and Probability Letters, 66, 315-325 (2004) · Zbl 1102.62054
[15] Wang, S., 1998. Aggregation of correlated risk portfolios: Models and algorithms. In: CAS Proceedings, pp. 848-939; Wang, S., 1998. Aggregation of correlated risk portfolios: Models and algorithms. In: CAS Proceedings, pp. 848-939
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