×

zbMATH — the first resource for mathematics

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.
Reviewer: Reviewer (Berlin)

MSC:
91B30 Risk theory, insurance (MSC2010)
60G40 Stopping times; optimal stopping problems; gambling theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.