Stability Bound of Ruin Probability in a Reduced Two-Dimensional Risk Model

Authors: Zina Benouaret, Djamil Aissani

Abstract:

In this work, we introduce the qualitative and quantitative concept of the strong stability method in the risk process modeling two lines of business of the same insurance company or an insurance and re-insurance companies that divide between them both claims and premiums with a certain proportion. The approach proposed is based on the identification of the ruin probability associate to the model considered, with a stationary distribution of a Markov random process called a reversed process. Our objective, after clarifying the condition and the perturbation domain of parameters, is to obtain the stability inequality of the ruin probability which is applied to estimate the approximation error of a model with disturbance parameters by the considered model. In the stability bound obtained, all constants are explicitly written.

Keywords: Markov chain, risk models, ruin probabilities, strong stability analysis.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1316853

References:

[1] D. Aissani, N. V. Kartashov, Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels. Compte Rendu Academy of Sciences U. S. S. R, ser. A, 11, 3-5, 1983.
[2] S. Asmussen and H. Albrecher, Ruin Probabilities. Vol. 14 of Advanced Series on Statistical Science Applied Probability, World Scientific, 2010.
[3] F. Avram, Z. Palmowski and M. Pistorius A two-dimensional ruin problem on the positive quadrant. Insurance: Mathematics and Economics 42 (1), 227-234, 2008.
[4] Z. Benouaret and D. Aissani, Strong stability in a two-dimensional classical risk model with independent claims. Compte Rendu Scandinavian Actuarial Journal. 2, 83-92, 2010.
[5] A. Touazi, Z. Benouaret, D. Aissani and S. Adjabi, Nonparametric estimation of the claim amount in the strong stability analysis of the classical risk model. Insurance: Mathematics and economics 74, 78-83, 2017.
[6] F. Enikeeva and V. Kalashnikov and D. Rusaityte, Continuity estimates for ruin probabilities. Scandinavian Actuarial Journal, 1, 18-39, 2001.
[7] V. Kalashnikov, The Stability concept for stochastic risk models. Working Paper Nr 166. Lab. of Actuarial Mathematics. University of Copenhagen, 2000.
[8] N. V. Kartashov, Strong Stable Markov Chains. VSP, Utrecht, 1999.
[9] D. Rusaityte,Stability bounds for ruin probabilities in a Markov modulated risk model with investments. Laboratory of Actuarial Mathematics, University of Copenhagen. Working Paper Nr. 178, 2001.
[10] V. Schmidt, T. Rolski, J. Teugels, and H. Schmidli, Stochastic Processes for Insurance and Finance, Wiley, 1999.