Survival probabilities in bivariate risk models, with application to reinsurance. (English) Zbl 1290.91077

Summary: This paper deals with an insurance portfolio that covers two interdependent risks. The central model is a discrete-time bivariate risk process with independent claim increments. A continuous-time version of compound Poisson type is also examined. Our main purpose is to develop a numerical method for determining non-ruin probabilities over a finite-time horizon. The approach relies on, and exploits, the existence of a special algebraic structure of Appell type. Some applications in reinsurance to the joint risks of the cedent and the reinsurer are presented and discussed, under a stop-loss or excess of loss contract.


91B30 Risk theory, insurance (MSC2010)
60G40 Stopping times; optimal stopping problems; gambling theory
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[1] Asmussen, S.; Albrecher, H., Ruin probabilities, (2010), World Scientific Singapore · Zbl 1247.91080
[2] Avram, F.; Palmowski, Z.; Pistorius, M., A two-dimensional ruin problem on the positive quadrant, Insurance: Mathematics & Economics, 42, 227-234, (2008) · Zbl 1141.91482
[3] Bowers, N. L.; Gerber, H. U.; Hickman, J. C.; Jones, D. A.; Nesbitt, C. J., Actuarial mathematics, (1997), The Society of Actuaries Schaumburg · Zbl 0634.62107
[4] Cai, J.; Li, H., Dependence properties and bounds for ruin probabilities in multivariate compound risk models, Journal of Multivariate Analysis, 98, 757-773, (2007) · Zbl 1280.91090
[5] Castañer, A.; Claramunt, M. M.; Gathy, M.; Lefèvre, C.; Mármol, M., Ruin problems for a discrete time risk model with non-homogeneous conditions, Scandinavian Actuarial Journal, 2013, 2, 83-102, (2013) · Zbl 1280.91091
[6] Castañer, A.; Claramunt, M. M.; Mármol, M., Ruin probability and time of ruin with a proportional reinsurance threshold strategy, TOP, 20, 614-638, (2012) · Zbl 1284.91211
[7] Centeno, M. L.; Simões, O., Optimal reinsurance, Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A-Matemáticas, 103, 387-404, (2009) · Zbl 1181.91090
[8] Chan, W.-S.; Yang, H.; Zhang, L., Some results on ruin probabilities in a two-dimensional risk model, Insurance: Mathematics & Economics, 32, 345-358, (2003) · Zbl 1055.91041
[9] Denuit, M.; Frostig, E.; Levikson, B., Supermodular comparison of time-to-ruin random vectors, Methodology and Computing in Applied Probability, 9, 41-54, (2007) · Zbl 1115.62111
[10] Denuit, M.; Lefèvre, C.; Picard, P., Polynomial structures in order statistics distributions, Journal of Statistical Planning and Inference, 113, 151-178, (2003) · Zbl 1031.62038
[11] De Vylder, F.; Goovaerts, M., Homogeneous risk models with equalized claim amounts, Insurance: Mathematics & Economics, 26, 223-238, (2000) · Zbl 1103.91361
[12] Dickson, D. C.; Waters, H. R., Reinsurance and ruin, Insurance: Mathematics & Economics, 19, 61-80, (1996) · Zbl 0894.62110
[13] Dimitrova, D. S.; Kaishev, V. K., Optimal joint survival reinsurance: an efficient frontier approach, Insurance: Mathematics & Economics, 47, 27-35, (2010) · Zbl 1231.91177
[14] Gong, L.; Badescu, A. L.; Cheung, E. C., Recursive methods for a multi-dimensional risk process with common shocks, Insurance: Mathematics & Economics, 50, 109-120, (2012) · Zbl 1235.91090
[15] Goovaerts, M. J.; Kaas, R.; Van Heerwaarden, A. E.; Bauwelinckx, T., Effective actuarial methods, (1990), North-Holland Amsterdam
[16] Guo, J. Y.; Yuen, K. C.; Zhou, M., Ruin probabilities in Cox risk models with two dependent classes of business, Acta Mathematica Sinica-English Series, 23, 1281-1288, (2007) · Zbl 1120.60069
[17] Ignatov, Z. G.; Kaishev, V. K., Two-sided bounds for the finite time probability of ruin, Scandinavian Actuarial Journal, 1, 46-62, (2000) · Zbl 0958.91030
[18] Ignatov, Z. G.; Kaishev, V. K., A finite-time ruin probability formula for continuous claim severities, Journal of Applied Probability, 41, 570-578, (2004) · Zbl 1048.60079
[19] Kaas, R., How to (and how not to) compute stop-loss premiums in practice, Insurance: Mathematics & Economics, 13, 241-254, (1993) · Zbl 0800.62681
[20] Kaas, R.; Goovaerts, M. J.; Dhaene, J.; Denuit, M., Modern actuarial risk theory: using R, (2008), Springer Heidelberg · Zbl 1148.91027
[21] Kaishev, V. K.; Dimitrova, D. S., Excess of loss reinsurance under joint survival optimality, Insurance: Mathematics & Economics, 39, 376-389, (2006) · Zbl 1151.91573
[22] Kaishev, V. K.; Dimitrova, D. S.; Ignatov, Z. G., Operational risk and insurance: a ruin-probabilistic reserving approach, Journal of Operational Risk, 3, 39-60, (2008)
[23] Kaz’min, Y. A., Appell polynomials, (Hazewinkel, M., Encyclopedia of Mathematics, Vol. 1, (1988), Kluwer Dordrecht), 209-210
[24] Lefèvre, C.; Loisel, S., Finite-time ruin probabilities for discrete, possibly dependent, claim severities, Methodology and Computing in Applied Probability, 11, 425-441, (2009) · Zbl 1170.91414
[25] Lefèvre, C.; Picard, P., A nonhomogeneous risk model for insurance, Computers and Mathematics with Applications, 51, 325-334, (2006) · Zbl 1161.91418
[26] Lefèvre, C.; Picard, P., A new look at the homogeneous risk model, Insurance: Mathematics & Economics, 49, 512-519, (2011) · Zbl 1229.91162
[27] Lefèvre, C.; Picard, P., Ruin probabilities for risk models with ordered claim arrivals, Methodology and Computing in Applied Probability, 1-23, (2013), (forthcoming)
[28] Li, J.; Liu, Z.; Tang, Q., On the ruin probabilities of a bidimensional perturbed risk model, Insurance: Mathematics & Economics, 41, 185-195, (2007) · Zbl 1119.91056
[29] Li, S.; Lu, Y.; Garrido, J., A review of discrete-time risk models, Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A-Matemáticas, 103, 321-337, (2009) · Zbl 1180.62151
[30] Mata, A. J., Pricing excess of loss reinsurance with reinstatements, Astin Bulletin, 30, 349-368, (2000) · Zbl 1060.91084
[31] Niederhausen, H., Sheffer polynomials, (Kotz, S.; Johnson, N.; Read, C., Encyclopedia of Statistical Sciences, Vol. 8, (1988), Wiley New York), 436-441
[32] Picard, P.; Lefèvre, C., First crossing of basic counting processes with lower non-linear boundaries: a unified approach through pseudopolynomials (i), Advances in Applied Probability, 28, 853-876, (1996) · Zbl 0857.60085
[33] Picard, P.; Lefèvre, C., The probability of ruin in finite time with discrete claim size distribution, Scandinavian Actuarial Journal, 1, 58-69, (1997) · Zbl 0926.62103
[34] Picard, P.; Lefèvre, C.; Coulibaly, I., Multirisks model and finite-time ruin probabilities, Methodology and Computing in Applied Probability, 5, 337-353, (2003) · Zbl 1035.62109
[35] Picard, P.; Lefèvre, C.; Coulibaly, I., Problèmes de ruine en théorie du risque à temps discret avec horizon fini, Journal of Applied Probability, 40, 527-542, (2003) · Zbl 1049.62113
[36] Rytgaard, M., Stop-loss reinsurance, (Teugels, J. L.; Sundt, B., Encyclopedia of Actuarial Science, Vol. 3, (2004), Wiley Chichester), 1620-1625
[37] Sangüesa, C., Error bounds in approximations of random sums using gamma-type operators, Insurance: Mathematics & Economics, 42, 484-491, (2008) · Zbl 1152.91601
[38] Sundt, B., On multivariate panjer recursions, Astin Bulletin, 29, 29-46, (1999)
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