Geodesic
Approximations of the ultimate ruin probability in the classical risk model using the Banach's fixed-point theorem and the continuity of the ruin probability
Kybernetika, Tome 58 (2022) no. 2, pp. 254-276
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In this paper, we show two applications of the Banach's Fixed-Point Theorem: first, to approximate the ultimate ruin probability in the classical risk model or Cramér-Lundberg model when claim sizes have some arbitrary continuous distribution and second, we propose an algorithm based in this theorem and some conditions to guarantee the continuity of the ruin probability with respect to the weak metric (Kantorovich). In risk theory literature, there is no methodology based in the Banach's Fixed-Point Theorem to calculate the ruin probability. Numerical results in this paper, guarantee a good approximation to the analytic solution of the ruin probability problem. Finally, we present numerical examples when claim sizes have distribution light and heavy-tailed.
In this paper, we show two applications of the Banach's Fixed-Point Theorem: first, to approximate the ultimate ruin probability in the classical risk model or Cramér-Lundberg model when claim sizes have some arbitrary continuous distribution and second, we propose an algorithm based in this theorem and some conditions to guarantee the continuity of the ruin probability with respect to the weak metric (Kantorovich). In risk theory literature, there is no methodology based in the Banach's Fixed-Point Theorem to calculate the ruin probability. Numerical results in this paper, guarantee a good approximation to the analytic solution of the ruin probability problem. Finally, we present numerical examples when claim sizes have distribution light and heavy-tailed.
DOI : 10.14736/kyb-2022-2-0254
Keywords: Banach's Fixed-Point Theorem; classical risk model; continuity of ruin probability; probabilistic metric; ultimate ruin probability. 
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Martínez Sánchez, Jaime; Baltazar-Larios, Fernando. Approximations of the ultimate ruin probability in the classical risk model using the Banach's fixed-point theorem and the continuity of the ruin probability. Kybernetika, Tome 58 (2022) no. 2, pp. 254-276. doi: 10.14736/kyb-2022-2-0254

[1] Asmussen, S., Binswanger, K.: Simulation of ruin probabilities for subexponential claims. ASTIN Bull. 27 (1997), 2, 297-318. | DOI

[2] Asmussen, S., Albrecher, H.: Ruin Probabilities. World Scientific Printers 2010. | MR

[3] Bladt, M., Nielsen, B. F., Samorodnitsky, G.: Calculation of ruin probabilities for a dense class of heavy-tailed distributions. Scand. Actuar. J. (2015), 573-591. | DOI | MR

[4] Bladt, M., Nielsen, B. F.: Matrix-exponential Distributions in Applied Probability. Springer, New York 2017. | MR

[5] Cai, J., Dickson, D. C. M.: Upper bounds for ultimate ruin probabilities in the Sparre Andersen model with interest. Insurance: Math. Econom. 32 (2002), 61-71. | DOI | MR

[6] Enikeeva, F., Kalashnikov, V., Rusaityle, D.: Continuity estimates of ruin probabilities. Scand. Actuar. J. 1 (2001), 18-39. | DOI | MR

[7] Gerber, H. U.: An Introducction to Mathematical Risk Theory. S. S. Huebner Foundation, Wharton School, Philadephia 1979. | MR

[8] Gerber, H., Shiu, E.: On the time value of ruin. North Amer. Actuar. J. 2 (1998), 48-72. | DOI | MR

[9] Gordienko, E., Vázquez-Ortega, P.: Simple continuity inequalities for ruin probability in the classical risk model. ASTIN Bull. 46 (2016), 801-814. | DOI | MR

[10] Granas, A., Dugundji, J.: Fixed Point Theory. New York, Springer-Verlag 2003. | MR

[11] Hernández-Lerma, O.: Adaptive Markov Control Processes. Springer-Verlag, New York 1989. | MR | Zbl

[12] Hernández-Lerma, O., Lasserre, J. Bernard: Discrete-Time Markov Control Processes. Basic Optimality Criteria. Springer, Berlin Heidelberg, New York 1996. | MR

[13] Kallenberg, O.: Probability and Its Applications. Second edition. Springer-Verlag, New York 2002. | MR

[14] Lee, SC, Lin, XS: Modeling and evaluating insurance losses via mixtures of Erlang distributions. North Amer. Actuar. J. 14 (2010), 1, 107-130. | DOI | MR

[15] Marceau, E., Rioux, J.: On robustness in risk theory. Insurance: Math. Econom. 29 (2001), 167-185. | DOI | MR

[16] Maciak, M., Okhrin, O., Pešta, M.: Infinitely stochastic micro reserving. Insurance: Math. Econom. 100 (2021), 30-58. | DOI | MR

[17] Panjer, H.: Direct calculation of ruin probabilities. J. Risk Insur. 53 (1986), 521-529. | DOI

[18] Rachev, S.: Probability Metrics and the Stability of Stochastic Models. John Wiley and Sons, 1981. | MR

[19] Rolski, T., Schmidli, H., Teugels, J.: Stochastic Processes for Insurance and Finance. John Wiley and Sons, 1999. | MR

[20] Ross, S., Schmidli, H.: Applied Probability Models with Optimization Applications. . Holden-Day, San Francisco 1970. | MR

[21] Santana, D., González-Hernández, J., Rincón, L.: Approximation of the ultimate ruin probability in the classical risk model using Erlang mixtures. Methodol. Comput. Appl. Probab. 19, (2017), 775-798. | DOI | MR

[22] Williams, D.: Probability its Martingale. Cambridge University Press, 1991. | MR

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