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On multivariate modifications of Cramer-Lundberg risk model with constant intensities. (English) Zbl 1411.62140

Summary: The paper considers very general multivariate modifications of Cramer-Lundberg risk model. The claims can be of different types and can arrive in groups. The groups arrival processes have constant intensities. The counting groups processes are dependent multivariate compound Poisson processes of Type I. We allow empty groups and show that in that case we can find stochastically equivalent Cramer-Lundberg model with non-empty groups. The investigated model generalizes the risk model with common shocks, the Poisson risk process of order \(k\), the Poisson negative binomial, the Polya-Aeppli, the Polya-Aeppli of order \(k\) among others. All of them with one or more types of policies. The numerical characteristics, Cramer-Lundberg approximations, and probabilities of ruin are derived. During the paper, we show that the theory of these risk models intrinsically relates to the special types of integro differential equations. The probability solutions to such differential equations provide new insights, typically overseen from the standard point of view.

MSC:

62H12 Estimation in multivariate analysis
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G10 Stationary stochastic processes
45J05 Integro-ordinary differential equations
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[1] Beekman, J. A., Collective risk results, Trans. Soc. Actuar., 20, 182-199 (1968)
[2] Chan, W.-S.; Yang, H.; Zhang, L., Some results on ruin probabilities in a two-dimensional risk model, Insur.: Math. Econ., 32, 3, 345-358 (2003) · Zbl 1055.91041
[3] Cossette, H.; Etienne, M., The discrete-time risk model with correlated classes of business, Insur.: Math. Econ., 26, 2, 133-149 (2000) · Zbl 1103.91358
[4] Cramér, H., On the Mathematical Theory of Risk (1930), Skandia Jubilee Volume. Stockholm, Sweden: Centraltryckeriet, Skandia Jubilee Volume. Stockholm, Sweden · JFM 56.1098.05
[5] Cramér, H., Collective Risk Theory: A Survey of the Theory From the Point of View of the Theory of Stochastic Processes (1955), Skandia Jubilee Volume. Stockholm, Sweden: Nordiska bokhandeln, Skandia Jubilee Volume. Stockholm, Sweden
[6] Chukova, S.; Minkova, L. D., Pólya-Aeppli of order k risk model, Commun. Stat.-Simul. Comput., 44, 3, 551-564 (2015) · Zbl 1334.60079
[7] Daley, D. J.; Vere-Jones, D., Basic properties of the Poisson process, Gani, J., Heyde, C. C., Kurtz, T. G., eds. An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods, New York, Berlin, Heidelberg, Hong Kong, London, Milan, Paris, Tokyo: Springer, 19-40 (2003)
[8] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling Extremal Events: For Insurance and Finance (1997), New York: Springer, New York · Zbl 0873.62116
[9] Gerber, H. U., An Introduction to Mathematical Risk Theory, 517/G36i. Berlin Heidelberg, New York: Springer, Swiss Association of Actuaries Zurich (1979)
[10] Gerber, H. U.; Shiu, E. S. W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insur.: Math. Econ., 21, 2, 129-137 (1997) · Zbl 0894.90047
[11] Goldie, C. M.; Klueppelberg, C. I., Subexponential distributions, Adler, R. J., Feldman, R. E., Taqqu, M. S., eds. A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Cambrige, MA, USA: Birkhauser Boston Inc, 435-459 (1998) · Zbl 0923.62021
[12] Grandell, J., Aspects of Risk Theory (1991), New York: Springer-Verlag, New York · Zbl 0717.62100
[13] Groparu-Cojocaru, I., A class of bivariate Erlang distributions and ruin probabilities in multivariate risk models, Ph.D. Thesis, Universite de Montreal (Canada) (2013)
[14] Johnson, N. L.; Kemp, A. W.; Kotz, S., Univariate Discrete Distributions (1992), New York: John Wiley & Sons, New York
[15] Khintchine, A. Y., Mathematical theory of a stationary queue, Matemat. Sbornik, 39, 4, 73-84 (1932)
[16] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models: From Data to Decisions, 715 (2012), New York: John Wiley & Sons, New York · Zbl 1272.62002
[17] Kostadinova, K., On a Poisson negative binomial process, Advanced Research in Mathematics and Computer Science, Doctoral Conference in Mathematics, Informatics and Education [MIE 2013] Proceedings, 25-33 (2013)
[18] Kostadinova, K., On the Poisson process of order k, Pliska Stud. Math. Bulgarica, 22, 1, 117-128 (2013) · Zbl 1374.60091
[19] Kostadinova, K., Polya-Aeppli risk model with two lines of business, Advanced Research in Mathematics and Computer Science, Doctoral Conference in Mathematics, Informatics and Education [MIE 2014] Proceedings, 27-31 (2014)
[20] Kostadinova, K.; Minkova, L., On a bivariate Poisson negative binomial risk process, Biomath, 3, 1, 47-52 (2014) · Zbl 1368.60039
[21] Lundberg, F., Some supplementary researches on the collective risk theory, Scand. Actuar. J., 15, 3, 137-158 (1932) · JFM 58.1177.04
[22] Marshall, A. W.; Ingram, O., A multivariate exponential distribution, J. Am. Stat. Assoc., 62, 317, 30-44 (1967) · Zbl 0147.38106
[23] Marshall, A. W.; Ingram, O., Families of multivariate distributions, J. Am. Stat. Assoc., 83, 403, 834-841 (1988) · Zbl 0683.62029
[24] Minkova, L. D., Compound compound Poisson risk model, Serdica Math. J., 35, 3, 301-310 (2009) · Zbl 1224.91078
[25] Minkova, L. D., The Pólya-Aeppli process and ruin problems, Int. J. Stoch. Anal., 3, 221-234 (2004) · Zbl 1127.91037
[26] Minkova, L. D.; Balakrishnan, N., Type II bivariate Pólya-Aeppli distribution, Stat. Probab. Lett., 88, 40-49 (2014) · Zbl 1294.60021
[27] Minkova, L. D.; Balakrishnan, N., On a bivariate Pólya-Aeppli distribution, Commun. Stat.-Theory Meth., 43, 23, 5026-5038 (2014) · Zbl 1307.60006
[28] Philippou, A. N., The Poisson and compound Poisson distributions of order k and some of their properties (1983) · Zbl 0529.60010
[29] Potocký, R., On a dividend strategy of insurance companies, Ekon. A Manag., 11, 4, 103-109 (2008)
[30] Potocký, R.; Waldl, H.; Stehlík, M., On sums of claims and their applications in analysis of pension funds and insurance products, Prague Econ. Papers, 3, 349-370 (2014)
[31] Pollaczek, F., Ber eine aufgabe der wahrscheinlichkeitstheorie. I, Math. Zeitschr., 32, 1, 64-100 (1930) · JFM 56.1087.01
[32] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J. L., Stochastic processes for insurance and finance, In: Barnett, V., Cressie, N. A. C., Fisher, N. I., Johnstone, I. M., Kadane, J. B., Kendall, D. G., Scott, D. W., Silverman, B. W., Smith, A. F. M., Teugels, J. L., (eds.), Bradley, R. A., Hunter, J. S., (Eds. Emeritus), Wiley series in probability and statistics. Chichester, New York, Weinheim, Brisbane, Singapore, Toronto: John Wiley and Sons (1998)
[33] Shiryayev, A. N., On analytical methods in probability theory, Selected Works of A. N. Kolmogorov. Mathematics and Its Applications (Soviet Series), 26, 62-108 (1992), Dordrecht: Springer, Dordrecht
[34] Smith, G. E. J., Parameter estimation in some multivariate compound distributions (1965)
[35] Wang, S., Aggregation of correlated risk portfolios: models and algorithms, Proc. Casual. Actuar. Soc., 85, 163, 848-939 (1998)
[36] Wang, G.; Yuen, K. C., On a correlated aggregate claims model with thinning-dependence structure, Insur.: Math. Econ., 36, 3, 456-468 (2005) · Zbl 1120.62095
[37] Yuen, K. C.; Xueyuan, W., On a correlated aggregate claims model with Poisson and Erlang risk processes, Insur.: Math. Econ., 31, 2, 205-214 (2002) · Zbl 1074.91566
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