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On the ruin probability of a generalized Cramér–Lundberg model driven by mixed Poisson processes

Part of: Applications Stochastic processes

Published online by Cambridge University Press:  12 September 2022

Masashi Tomita ,
Koichiro Takaoka  and
Motokazu Ishizaka Open the ORCID record for Motokazu Ishizaka [Opens in a new window]
Masashi Tomita*
Affiliation:
Meiji Yasuda Life Insurance Company
Koichiro Takaoka*
Affiliation:
Chuo University
Motokazu Ishizaka*
Affiliation:
Chuo University
*
*Postal address: 1-1, Marunouchi 2-chome, Chiyoda-ku, Tokyo 100-0005, Japan. Email: ma-tomita@meijiyasuda.co.jp
**Postal address: Faculty of Commerce, Chuo University, 742-1 Higashinakano, Hachioji-shi, Tokyo 192-0393, Japan.
***Email address: takaoka@tamacc.chuo-u.ac.jp
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Abstract

We propose a generalized Cramér–Lundberg model of the risk theory of non-life insurance and study its ruin probability. Our model is an extension of that of Dubey (1977) to the case of multiple insureds, where the counting process is a mixed Poisson process and the continuously varying premium rate is determined by a Bayesian rule on the number of claims. We use two proofs to show that, for each fixed value of the safety loading, the ruin probability is the same as that of the classical Cramér–Lundberg model and does not depend on either the distribution of the mixing variable of the driving mixed Poisson process or the number of claim contracts.

Keywords

Risk theoryvarying insurance premiumconditional distributionBayesian estimatoradjustment coefficient

MSC classification

Secondary: 60G44: Martingales with continuous parameter 62P05: Applications to actuarial sciences and financial mathematics
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 849 - 859
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Afonso, L. B. (2008). Evaluation of ruin probabilities for surplus processes with credibility and surplus dependent premiums. PhD thesis, ISEG, Lisbon.Google Scholar
Afonso, L. B., dos Reis, A. D. E. and Waters, H. R. (2010). Numerical evaluation of continuous-time ruin probabilities for a portfolio with credibility updated premiums. ASTIN Bull. 40, 399–414.10.2143/AST.40.1.2049236CrossRefGoogle Scholar
Asmussen, S. (1999). On the ruin problem for some adapted premium rules. In Probabilistic Analysis of Rare Events, eds V. K. Kalashnikov and A. M. Andronov, Riga Aviation University, pp. 3–15.Google Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Singapore.10.1142/7431CrossRefGoogle Scholar
Boudreault, M., Cossette, H., Landriault, D. and Marceau, E. (2006). On a risk model with dependence between interclaim arrivals and claim sizes. Scand. Actuarial J. 2006, 265–285.10.1080/03461230600992266CrossRefGoogle Scholar
Cossette, H. and Marceau, E. (2000). The discrete-time risk model with correlated classes of business. Insurance Math. Econom. 26, 133–149.10.1016/S0167-6687(99)00057-8CrossRefGoogle Scholar
Dubey, A. (1977). Probabilité de ruine lorsque le paramètre de Poisson est ajusté a posteriori. Mitt. Ver. Schweiz. Versicherungsmath. 2, 130–141.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J. 2, 48–72.10.1080/10920277.1998.10595671CrossRefGoogle Scholar
Grandell, J. (1997). Mixed Poisson Processes. Chapman and Hall, London.10.1007/978-1-4899-3117-7CrossRefGoogle Scholar
Grigelionis, B. (1977). Martingale characterization of stochastic processes with independent increments. Litovsk. Math. Sb. 17, 75–86.Google Scholar
Igarashi, T. (2015). When to reduce insurance premium? An optimal stopping approach. Masters thesis, Hitotsubashi University, Tokyo (in Japanese).Google Scholar
Kallenberg, O. (2017). Random Measures, Theory and Applications. Springer, Berlin.10.1007/978-3-319-41598-7CrossRefGoogle Scholar
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012). Loss Models: From Data to Decisions, 4th edn. John Wiley and Sons, Chichester.Google Scholar
Landriault, D., Lemieux, C. and Willmot, G. E. (2012). An adaptive premium policy with a Bayesian motivation in the classical risk model. Insurance Math. Econom. 51, 370–378.10.1016/j.insmatheco.2012.06.001CrossRefGoogle Scholar
Li, S., Landriault, D. and Lemieux, C. (2015). A risk model with varying premiums: Its risk management implications. Insurance Math. Econom. 60, 38–46.10.1016/j.insmatheco.2014.10.010CrossRefGoogle Scholar
Lundberg, F. (1903). Approximerad framställning av sannolikhetsfunktionen. Återförsäkring av kollektivrisker. Acad. Afhaddling. Almqvist. och Wiksell, Uppsala.Google Scholar
Minkova, L. D. (2004). The Pólya–Aeppli process and ruin problems. J. Appl. Math. Stoch. Anal. 2004, 221–234.10.1155/S1048953304309032CrossRefGoogle Scholar
Müller, A., and Pflug, G. (2001). Asymptotic ruin probabilities for risk processes with dependent increments. Insurance Math. Econom. 28, 381–392.10.1016/S0167-6687(01)00063-4CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. L. (1998). Stochastic Process for Insurance and Finance. John Wiley and Sons, Chichester.Google Scholar
Trufin, J., and Loisel, S. (2013). Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments. Bulletin Français d’Actuariat 13, 73–102.Google Scholar
Tsai, C., and Parker, G. (2004). Ruin probabilities: Classical versus credibility. In Proc. NTU Int. Conf. Finance.Google Scholar