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On finite-time ruin probabilities for classical risk models. (English) Zbl 1164.91033
The aim of this paper is to establish that the formula derived by Picard and Lefèvre can be generalized to the classical compound binomial and compound Poisson risk models. This extended formula is given explicitly in terms of the pseudo-distributions of some cumulated claim amounts. To begin, the authors obtain a first formula for the finite-time probability of non-ruin by applying a standard argument based on the ballot theorem when the initial reserves are zero, and on a Seal-type formula when the reserves are positive. The authors introduce a concept of pseudo-distribution for the cumulated claim amounts, and it is shown how to numerically compute these pseudo-laws using simple recursions. Using these results, the authors obtain the desired Picard and Lefèvre formula. Then two expressions for the (non)-ruin probability over an infinite horizon are obtained.

MSC:
91B30 Risk theory, insurance (MSC2010)
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[1] DOI: 10.1016/S0167-6687(01)00080-4 · Zbl 1025.62036 · doi:10.1016/S0167-6687(01)00080-4
[2] DOI: 10.1142/9789812779311 · doi:10.1142/9789812779311
[3] Cardoso R. M. R., Insurance: Mathematics and Economics 33 pp 659– (2003) · Zbl 1103.60314 · doi:10.1016/j.insmatheco.2003.09.008
[4] DOI: 10.1016/S0167-6687(99)00053-0 · Zbl 1013.91063 · doi:10.1016/S0167-6687(99)00053-0
[5] Cramér H., On the mathematical theory of risk (1930) · JFM 56.1100.03
[6] De Vylder F. E., Bulletin Français d’ Actuariat 1 pp 31– (1997)
[7] De Vylder F. E., Scandinavian Actuarial Journal 2 pp 97– (1999) · Zbl 0952.91042 · doi:10.1080/03461239950132598
[8] De Vylder F. E., Scandinavian Actuarial Journal 2 pp 109– (1996) · Zbl 0880.62108 · doi:10.1080/03461238.1996.10413967
[9] DOI: 10.1016/0167-6687(88)90089-3 · Zbl 0629.62101 · doi:10.1016/0167-6687(88)90089-3
[10] DOI: 10.1016/S0167-6687(98)00049-3 · Zbl 0963.91062 · doi:10.1016/S0167-6687(98)00049-3
[11] Dickson D. C. M., British Actuarial Journal 5 pp 575– (1999) · doi:10.1017/S135732170000057X
[12] Gerber H. U., An introduction to mathematical risk theory (1979) · Zbl 0431.62066
[13] DOI: 10.2143/AST.18.2.2014949 · doi:10.2143/AST.18.2.2014949
[14] Gerber H. U., North American Actuarial Journal 2 pp 48– (1998) · Zbl 1081.60550 · doi:10.1080/10920277.1998.10595671
[15] Grandell J., Aspects of risk theory (1990) · Zbl 0717.62100
[16] Grimmett G. R., Probability and stochastic processes (1992) · Zbl 0759.60002
[17] DOI: 10.1080/034612300750066728 · Zbl 0958.91030 · doi:10.1080/034612300750066728
[18] DOI: 10.1239/jap/1082999087 · Zbl 1048.60079 · doi:10.1239/jap/1082999087
[19] DOI: 10.1016/S0167-6687(01)00078-6 · Zbl 1074.62528 · doi:10.1016/S0167-6687(01)00078-6
[20] Kaas R., Modern actuarial risk theory (2001) · Zbl 1086.91035
[21] Konstantopoulos T., Statistics and Probability Letters 24 pp 331– (1995) · Zbl 0832.60017 · doi:10.1016/0167-7152(94)00191-A
[22] Li , S. & Garrido , J. ( 2002 ). On the time value of ruin in the discrete time risk model . Working paper 02-18, Business Economics, University Carlos III of Madrid .
[23] Loisel , St. , Mazza , Ch. & Rullière , D. ( 2007 ). Robustness analysis, convergence of empirical finite-time ruin probabilities and estimation of risk solvency margin . Working paper, Cahiers de recherche de l’I.S.F.A., Université Lyon 1 .
[24] Panjer H. H., Insurance risk models (1992)
[25] Picard Ph., Scandinavian Actuarial Journal 1 pp 58– (1997)
[26] Picard Ph., Journal of Applied Probability 40 pp 543– (2003) · Zbl 1140.91408 · doi:10.1239/jap/1059060887
[27] Picard Ph., Journal of Applied Probability 40 pp 527– (2003) · Zbl 1049.62113 · doi:10.1239/jap/1059060886
[28] DOI: 10.1214/aoms/1177704970 · Zbl 0103.13302 · doi:10.1214/aoms/1177704970
[29] Rullière D., Insurance: Mathematics and Economics 35 pp 187– (2004) · Zbl 1103.91048 · doi:10.1016/j.insmatheco.2004.07.001
[30] Seal H. L., The stochastic theory of a risk business (1969) · Zbl 0196.23501
[31] Shiu E. S. W., Insurance: Mathematics and Economics 7 pp 41– (1988) · Zbl 0664.62112 · doi:10.1016/0167-6687(88)90095-9
[32] DOI: 10.2143/AST.19.2.2014907 · doi:10.2143/AST.19.2.2014907
[33] DOI: 10.1017/S0269964803174025 · Zbl 1053.60049 · doi:10.1017/S0269964803174025
[34] DOI: 10.2307/2281642 · Zbl 0109.36702 · doi:10.2307/2281642
[35] Takács L., Combinatorial methods in the theory of stochastic processes (1967)
[36] DOI: 10.1016/0167-6687(93)90823-8 · Zbl 0778.62099 · doi:10.1016/0167-6687(93)90823-8
[37] DOI: 10.1016/j.jspi.2003.05.003 · Zbl 1073.60045 · doi:10.1016/j.jspi.2003.05.003
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