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SIMPLE CONTINUITY INEQUALITIES FOR RUIN PROBABILITY IN THE CLASSICAL RISK MODEL

Published online by Cambridge University Press:  05 May 2016

Evgueni Gordienko  and
Patricia Vázquez-Ortega
Evgueni Gordienko
Affiliation:
Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340 Iztapalapa, MéxicoD.F. E-mail: [email protected]
Patricia Vázquez-Ortega*
Affiliation:
Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340 Iztapalapa, MéxicoD.F. E-mail: [email protected]
*
E-mail: [email protected]
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Abstract

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A simple technique for continuity estimation for ruin probability in the compound Poisson risk model is proposed. The approach is based on the contractive properties of operators involved in the integral equations for the ruin probabilities. The corresponding continuity inequalities are expressed in terms of the Kantorovich and weighted Kantorovich distances between distribution functions of claims. Both general and light-tailed distributions are considered.

Keywords

Risk modelinfinite-time ruin probabilitycontinuity inequalitiescontractive operatorsKantorovich's metric
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 46 , Issue 3 , September 2016 , pp. 801 - 814
Copyright
Copyright © Astin Bulletin 2016 

References

Asmussen, S. and Albrecher, H. (2010) Ruin Probabilities. Singapore: World Scientific Printers.Google Scholar
Bening, V.E. and Korolev, V. (2003) Nonparametric estimation of the ruin probability for generalized risk processes. Theory of Probability and Its Applications, 47, 1–16.Google Scholar
Benouaret, Z. and Aïssani, D. (2010) Strong stability in a two-dimensional classical risk model with independent claims. Scandinavian Actuarial Journal, 2, 83–92.Google Scholar
Bobkov, S. and Ledoux, M. (2014) One-dimensional empirical measures, order statistics and Kantorovich transport distance. Preprint. University of Minessota and University of Toulouse. perso.math.univ-toulouse.fr/ledoux/files/2014/04/Order.statistics.pdf.Google Scholar
Cai, J. and Dickson, D.C.M. (2002) Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest. Insurance: Mathematics and Economics, 32, 61–71.Google Scholar
Chan, G. and Yang, H. (2005) Sensitivity analysis on ruin probabilities with heavy-tailed claims. Statistical Methodology, 2, 59–63.CrossRefGoogle Scholar
Enikeeva, F., Kalashnikov, V. and Rusaityte, D. (2001) Continuity estimates for ruin probabilities. Scandinavian Actuarial Journal, 1, 18–39.CrossRefGoogle Scholar
Frees, E. (1986) Nonparametric estimation of the probability of ruin. ASTIN Bulletin, 16, 81–92.Google Scholar
Kalashnikov, V.V. (1996) Two-sided bounds of ruin probability. Scandinavian Actuarial Journal, 1, 1–18.CrossRefGoogle Scholar
Kalashnikov, V.V. (1997) Geometric Sums: Bounds for Rare Events with Applications. Dordrecht: Kluwer Academic Publishers.Google Scholar
Kalashnikov, V.V. (2000) The stability concept for stochastic risk models. Working Paper No. 166. Laboratory of Actuarial Mathematics, University of Copenhagen.Google Scholar
Kass, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory. Boston: Kluwer Academic Publishers.Google Scholar
Kartashov, N.V. (1986) Inequalities in theorems of ergodicity and stability for Markov Chains with common phase space, II. Theory of Probability and Its Applications, 30, 507–515.Google Scholar
Kartashov, N.V. (2001) Interval estimates of the classical ruin probability for some classes of distributions of claims. Theory of Probability and Mathematical Statistics, 63, 87–97.Google Scholar
Marceau, E. and Rioux, J. (2001) On robustness in risk theory. Insurance: Mathematics and Economics, 29, 167–185.Google Scholar
Meyn, S. and Tweedie, R. (1993) Markov Chains and Stochastic Stability. New York: Springer.Google Scholar
Mnatsakanov, R., Ruymgaart, L.L. and Ruymgaart, F.H. (2008) Nonparametric estimation of ruin probabilities given a random sample of claims. Mathematical Methods of Statistics, 17, 35–43.CrossRefGoogle Scholar
Rachev, S.T. (1991) Probability Metrics and Stability of Stochastic Models. Chichester: John Wiley & Sons.Google Scholar
Rachev, S.T. and Rüschendorf, L. (1998) Mass Transportation Problems. Vol. I: Theory. New York: Springer.Google Scholar
Rolski, T., Schmidli, H. and Teugels, J. (1999) Stochastic Processes for Insurance and Finance. England: John Wiley and Sons.Google Scholar
Rusaityte, D. (2001) Stability bounds for ruin probabilities in a Markov modulated risk model with investments. Working Paper No. 178. Laboratory of actuarial mathematics, University of Copenhagen.Google Scholar
Yu, M.A. (2005) Sensitivity and convergence of uniformly ergodic Markov chains. Journal of Applied Probabilities, 42, 1003–1014.Google Scholar
Yuanjiang, H., Xucheng, L. and Zhang, J. (2003) Some results of ruin probability for the classical risk process. Journal of Applied Mathematics and Decision Sciences, 7 (3), 133–146.Google Scholar