×

On an asymptotic rule \(A+B/u\) for ultimate ruin probabilities under dependence by mixing. (English) Zbl 1290.91084

Summary: The purpose of this paper is to point out that an asymptotic rule \(A+B/u\) for the ultimate ruin probability applies to a wide class of dependent risk processes, in continuous or discrete time. That dependence is incorporated through a mixing model in the individual claim amount distributions. Several special mixing distributions are examined in detail and some close-form formulas are derived. Claim tail distributions and the dependence structure are also investigated.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62H05 Characterization and structure theory for multivariate probability distributions; copulas

Software:

gmp; SageMath; MPFR; R; DLMF
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albrecher, H.; Asmussen, S., Ruin probabilities and aggregate claims distributions for shot noise Cox processes, Scandinavian Actuarial Journal, 2006, 2, 86-110 (2006) · Zbl 1129.91022
[2] Albrecher, H.; Boxma, O., A ruin model with dependence between claim sizes and claim intervals, Insurance: Mathematics & Economics, 35, 2, 245-254 (2004) · Zbl 1079.91048
[3] Albrecher, H.; Constantinescu, C.; Loisel, S., Explicit ruin formulas for models with dependence among risks, Insurance: Mathematics & Economics, 48, 2, 265-270 (2011) · Zbl 1218.91065
[4] Albrecher, H.; Teugels, J. L., Exponential behavior in the presence of dependence in risk theory, Journal of Applied Probability, 43, 1, 265-285 (2006)
[6] Asmussen, S.; Albrecher, H., Ruin Probabilities (2010), World Scientific · Zbl 1247.91080
[7] Asmussen, S.; Rolski, T., Computational methods in risk theory: a matrix algorithmic approach, Insurance: Mathematics & Economics, 10, 4, 259-274 (1991) · Zbl 0748.62058
[8] Bak, J.; Newman, D. J., Complex Analysis (2010), Springer · Zbl 1205.30001
[9] Boudreault, M.; Cossette, H.; Landriault, D.; Marceau, E., On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian Actuarial Journal, 2006, 5, 265-285 (2006) · Zbl 1145.91030
[10] Cai, J.; Li, H., Multivariate risk model of phase type, Insurance: Mathematics & Economics, 36, 2, 137-152 (2005) · Zbl 1122.60049
[11] Centeno, M. D.L., Excess of loss reinsurance and Gerber’s inequality in the Sparre Anderson model, Insurance: Mathematics & Economics, 31, 3, 415-427 (2002) · Zbl 1074.91567
[12] Chaudry, M. A.; Zubair, S. M., On a Class of Incomplete Gamma Functions with Applications (2002), Chapman and Hall
[13] Clarke, F. H.; Bessis, D. N., Partial subdifferentials, derivates and Rademacher’s theorem, Transactions of the American Mathematical Society, 351, 7, 2899-2926 (1999) · Zbl 0924.49013
[14] Collamore, J., Hitting probabilities and large deviations, The Annals of Probability, 24, 4, 2065-2078 (1996) · Zbl 0879.60021
[15] Constantinescu, C.; Hashorva, E.; Ji, L., Archimedean copulas in finite and infinite dimensions—with application to ruin problems, Insurance: Mathematics & Economics, 49, 3, 487-495 (2011) · Zbl 1284.62339
[16] Dimitrova, D. S.; Kaishev, V. K., Optimal joint survival reinsurance: an efficient frontier approach, Insurance: Mathematics & Economics, 47, 1, 27-35 (2010) · Zbl 1231.91177
[17] Embrechts, P.; Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: Mathematics & Economics, 1, 1, 55-72 (1982) · Zbl 0518.62083
[18] Flajolet, P.; Sedgewick, R., Mellin transforms and asymptotics: finite differences and Rice’s integrals, Theoretical Computer Science, 144, 1-2, 101-124 (1995) · Zbl 0869.68056
[20] Frees, E. W.; Wang, P., Copula credibility for aggregate loss models, Insurance: Mathematics & Economics, 38, 360-373 (2006) · Zbl 1132.91489
[21] Genest, C.; Nešlehová, J., A primer on copulas for count data, ASTIN Bulletin, 37, 2, 475-515 (2007) · Zbl 1274.62398
[22] Gerber, H. U., Mathematical fun with compound binomial process, ASTIN Bulletin, 18, 2, 161-168 (1988)
[23] Gerber, H. U.; Shiu, E. S., On the time value of ruin, North American Actuarial Journal, 2, 1, 48-78 (1998) · Zbl 1081.60550
[24] Gerber, H. U.; Shiu, E. S., The time value of ruin in a Sparre Andersen model, North American Actuarial Journal, 9, 2, 49-84 (2005) · Zbl 1085.62508
[25] Gordon, R. A., The Integrals of Lebesgue, Denjoy, Perron and Henstock, Vol. 4 (1994), American Mathematical Society · Zbl 0807.26004
[27] Hildebrandt, T., Introduction to the Theory of Integration (1971), Routledge
[28] Huang, H.-N.; Marcantognini, S.; Young, N., Chain rules for higher derivatives, The Mathematical Intelligencer, 28, 2, 61-69 (2006) · Zbl 1172.46027
[29] Ignatov, Z. G.; Kaishev, V. K., Finite time non-ruin probability for Erlang claim inter-arrivals and continuous interdependent claim amounts, Stochastics: An International Journal of Probability and Stochastic Processes, 84, 4, 461-485 (2012) · Zbl 1262.91094
[30] Jeffrey, A.; Dai, H.-H., Handbook of Mathematical Formulas and Integrals (2008), Academic Press · Zbl 1139.00004
[31] Joe, H., Multivariate dependence measure and data analysis, (Monographs on Statistics and Applied Probability, Vol. 73 (1997), Chapman and Hall)
[32] Jones, D. S., Introduction to Asymptotics: a Treatment using Nonstandard Analysis (1997), World Scientific · Zbl 0915.34001
[33] Klüppelberg, C.; Stadtmüller, U., Ruin probabilities in the presence of heavy-tails and interest rates, Scandinavian Actuarial Journal, 1998, 1, 49-58 (1998) · Zbl 1022.60083
[34] Lefèvre, C.; Loisel, S., On finite-time ruin probabilities for classical risk models, Scandinavian Actuarial Journal, 2008, 1, 41-60 (2008) · Zbl 1164.91033
[35] Lefèvre, C.; Loisel, S., Finite-time ruin probabilities for discrete, possibly dependent, claim severities, Methodology and Computing in Applied Probability, 11, 3, 425-441 (2009) · Zbl 1170.91414
[36] Lefèvre, C.; Picard, P., A new look at the homogeneous risk model, Insurance: Mathematics & Economics, 49, 3, 512-519 (2011) · Zbl 1229.91162
[37] Li, S.; Lu, Y.; Garrido, J., A review of discrete-time risk models, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 103, 2, 321-337 (2009) · Zbl 1180.62151
[38] Lu, Y.; Garrido, J., Doubly periodic non-homogeneous Poisson models for hurricane data, Statistical Methodology, 2, 1, 17-35 (2005) · Zbl 1248.86003
[39] Maechler, M., Rmpfr: R MPFR—Multiple Precision Floating-Point Reliable (2012), ETH Zurich
[40] Merentes, N., On the composition operator in AC[a,b], Collectanea Mathematica, 42, 3, 237-243 (1991) · Zbl 0783.47045
[41] Nelsen, R. B., An Introduction to Copulas (2006), Springer · Zbl 1152.62030
[42] Nolan, J. P., Stable Distributions: Models for Heavy Tailed Data (2012), Birkhāuser
[43] (Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Cambridge University Press), URL: http://dlmf.nist.gov/ · Zbl 1198.00002
[44] Picard, P.; Lefèvre, C., Probabilité de ruine éventuelle dans un modèle de risque à temps discret, Journal of Applied Probability, 40, 3, 543-556 (2003) · Zbl 1140.91408
[45] Picard, P.; Lefèvre, C.; Coulibaly, I., Multirisks model and finite-time ruin probabilities, Methodology and Computing in Applied Probability, 5, 3, 337-353 (2003) · Zbl 1035.62109
[47] Reiss, R.-D.; Thomas, M., Statistical Analysis of Extreme Values (2007), Birkhāuser
[48] Shiu, E. S.W., The probability of eventual ruin in the compound binomial model, ASTIN Bulletin, 19, 2, 179-190 (1989)
[49] Simon, H. A., On a class of skew distribution functions, Biometrika, 42, 3/4, 425-440 (1955) · Zbl 0066.11201
[50] Song, M.; Meng, Q.; Wu, R.; Ren, J., The Gerber-Shiu discounted penalty function in the risk process with phase-type interclaim times, Applied Mathematics and Computation, 216, 2, 523-531 (2010) · Zbl 1202.91129
[52] Sundt, B.; dos Reis, A. D.E., Cramér-Lundberg results for the infinite time ruin probability in the compound binomial model, Bulletin of the Swiss Association of Actuaries, 2, 179-190 (2007) · Zbl 1333.91038
[53] Trufin, J.; Loisel, S., Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments, Bulletin Français d’Actuariat, 13, 25, 73-102 (2013)
[54] Willmot, G. E., Ruin probabilities in the compound binomial model, Insurance: Mathematics & Economics, 12, 2, 133-142 (1993) · Zbl 0778.62099
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.