Abstract
We consider the problem of approximating the infinite time horizon ruin probabilities for discrete time risk processes. The approach is based on asymptotic results for non-linearly perturbed discrete time renewal equations. Under some moment conditions on the claim distributions, the approximations take the form of exponential asymptotic expansions with respect to the perturbation parameter. We show explicitly how the coefficients of these expansions can be computed as functions of the coefficients of the expansions of local characteristics for perturbed risk processes.
Keywords
- Discrete Time Risk Model
- Ruin Probability
- Perturbed Risk Process
- Exponential Asymptotic Expansion
- Silvestrov
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Blanchet, J., Zwart, B.: Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processors sharing queues. Math. Methods Oper. Res. 72(2), 311–326 (2010)
Cossette, H., Landriault, D., Marceau, E.: Compound binomial risk model in a markovian environment. Insur. Math. Econ. 35, 425–443 (2004)
Dickson, D.C.M., Egídio dos Reis, A.D., Waters, H.R.: Some stable algorithms in ruin theory and their applications. Astin Bull. 25(2), 153–175 (1995)
Englund, E., Silvestrov, D.S.: Mixed large deviation and ergodic theorems for regenerative processes with discrete time. In: Jagers P., Kulldorff G., Portenko N., Silvestrov D. (eds.) Proceedings of the Second Scandinavian-Ukrainian Conference in Mathematical Statistics, vol. I, Umeå (1997) Theory stochastic Process, vol 3(19), no. 1–2, pp. 164–176 (1997)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. II. Wiley Series in Probability and Statistics. Wiley, New York (1966, 1971)
Gerber, H.U.: Mathematical fun with the compound binomial process. Astin Bull. 18(2), 161–168 (1988)
Grandell, J.: Aspects of Risk Theory. Probability and its Applications. Springer, New York (1991)
Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary distributions of a stochastic metapopulation model. J. Math. Biol. 33, 35–70 (1994)
Gyllenberg, M., Silvestrov, D.S.: Cramér-Lundberg approximation for nonlinearly perturbed risk processes. Insur. Math. Econ. 26, 75–90 (2000)
Gyllenberg, M., Silvestrov, D.S.: Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems. De Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin (2008)
Kartashov, M.V.: Inhomogeneous perturbations of a renewal equation and the Cramér-Lundberg theorem for a risk process with variable premium rates. Theory Probab. Math. Stat. 78, 61–73 (2009)
Li, S., Lu, Y., Garrido, J.: A review of discrete-time risk models. Rev. R. Acad. Cien. Serie A. Mat. 103(2), 321–337 (2009)
Ni, Y.: Perturbed renewal equations with multivariate non-polynomial perturbations. In: Frenkel I., Gertsbakh I., Khvatskin L., Laslo Z., Lisnianski A. (eds.) Proceedings of the International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management, pp. 754–763. Beer Sheva, Israel (2010a)
Ni, Y.: Analytical and numerical studies of perturbed renewal equations with multivariate non-polynomial perturbations. J. Appl. Quant. Methods 5(3), 498–515 (2010b)
Petersson, M.: Quasi-stationary distributions for perturbed discrete time regenerative processes. Theory Probab. Math. Stat. 89 (2013, forthcoming)
Petersson, M., Silvestrov, D.: Asymptotic expansions for perturbed discrete time renewal equations and regenerative processes. Research Report 2012:12, Department of Mathematics, Stockholm University, 34 pp. (2012)
Shiu, E.S.W.: The probability of eventual ruin in the compound binomial model. Astin Bull. 19(2), 179–190 (1989)
Silvestrov, D., Petersson, M.: Exponential expansions for perturbed discrete time renewal equations. In: Frenkel, I., Karagrigoriou, A., Lisnianski, A., Kleyner, A. (eds.) Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference, pp. 349–362. Wiley, Chichester (2014)
Willmot, G.E.: Ruin probabilities in the compound binomial model. Insur. Math. Econ. 12, 133–142 (1993)
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Petersson, M. (2014). Asymptotics of Ruin Probabilities for Perturbed Discrete Time Risk Processes. In: Silvestrov, D., Martin-Löf, A. (eds) Modern Problems in Insurance Mathematics. EAA Series. Springer, Cham. https://doi.org/10.1007/978-3-319-06653-0_7
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DOI: https://doi.org/10.1007/978-3-319-06653-0_7
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