Abstract
The paper overviews stochastic optimization models of actuarial mathematics and methods for their solution from the point of view of the methodology of multicriteria stochastic programming and optimal control. The evolution of the capital of an insurance company is considered in discrete time. The main random parameters of the models are insurance payouts, i.e., the ratios of paid insurance claims to the corresponding premiums per unit time. Optimization variables are the structure of the insurance portfolio (gross premium structure) and amount of dividends. As efficiency criteria, indicators of the profitability of the insurance business are used, and, as risk indicators the ruin probability and the recourse capital necessary to prevent the ruin are taken. The goal of the optimization is to find Pareto-optimal solutions. Methods for finding these solutions are proposed.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
R. Kaas, M. Goovaerts, J. Dhaene, and M. Denuit, Modern Actuarial Risk Theory, Kluwer Academic Publ., Boston (2001).
N. L. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones, and C. J. Nesbitt, Actuarial Mathematics, Society of Actuaries, Chicago (1997).
B. de Finetti, “Su un’impostazione alternativa della teoria collettiva del rischio,” in: Trans. 15th Intern. Congress of Actuaries, Vol. 2, Actuarial Society of America, New York (1957), pp. 433–443.
R. E. Beard, T. Pentikainen, and E. Pesonen, Risk Theory. The Stochastic Basis of Insurance, Chapman and Hall,London–New York (1984).
A. N. Nakonechnyi, “Optimization of risk processes,” Cybern. Syst. Analysis, Vol. 32, No. 5, 641–646 (1996).
G. I. Lyubchenko and A. N. Nakonechnyi, “Optimization methods for compound Poisson risk processes,” Cybern. Syst. Analysis, Vol. 34, No. 2, 230–237 (1998).
E. Bayraktar and V. Young, “Maximizing utility of consumption subject to a constraint on the probability of lifetime ruin,” Finance and Research Letters, Vol. 5, Iss. 4, 204–212 (2008). DOI: https://doi.org/10.1016/j.frl.2008.08.002.
H. Schmidli, Stochastic Control in Insurance, Springer-Verlag, London (2008).
B. V. Norkin, “On optimization of insurance portfolio,” Applied Statistics. Actuarial and Financial Mathematics, No. 1–2, 197–203 (2011). URL: https://www.researchgate.net/publication/260869967_On_optimization_of_insurance_portfolio.
B. V. Norkin, “Mathematical models for insurance business optimization,” Cybern. Syst. Analysis, Vol. 47, No. 1, 117–133 (2011).
A. N. Gromov, “Optimal reinsurance and investment strategy,” Vestn. Mosk. Univer., Ser. 1, Mathematics. Mechanics, Issue 2, 6–12 (2013). URL: https://cyberleninka.ru/article/v/optimalnaya-strategiya-perestrahovaniyai-investirovaniya.
S. Asmussen and H. Albrecher, Ruin Probabilities, World Scientific, London (2010).
V. Yu. Korolev, B. E. Bening, and S. Ya. Shorgin, The Mathematical Fundamentals of Risk Theory [in Russian], Fizmatlit, Moscow (2011).
M. M. Leonenko, Yu. S. Mishura, Ya. M. Parkhomenko, and M. J. Yadrenko, Probability-Theoretic and Statistical Methods in Econometry and Financial Mathematics [in Ukrainian], Informtekhnika, Kyiv (1995).
A. Shapiro, D. Dentcheva, and A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia (2009).
Yu. M. Ermoliev, Methods of Stochastic Programming [in Russian], Nauka, Moscow (1976).
Yu. S. Kan and A. N. Kibzun, Stochastic Programming Problems with Probability Criteria [in Russian], Fizmatlit, Moscow (2009).
D. B. Yudin, Problems and Methods of Stochastic Programming [in Russian], Sov. Radio, Moscow (1979).
W. B. Powell, “Clearing the jungle of stochastic optimization,” in: INFORMS Tutorials in Operations Research, Published online: 27 Oct. (2014), pp. 109–137. DOIhttps://doi.org/10.1287/educ.2014.0128.
B. V. Norkin, Numerical Methods to Solve Stochastic Optimization Problems in Actuarial Mathematics, Author’s Abstracts of Ph.D. Theses, V. M. Glushkov Institute of Cybernetics, NAS of Ukraine (2015).
Forinsurer. URL: https://forinsurer.com/ratings/nonlife.
Y. M. Ermoliev, T. Y. Ermolieva, G. MacDonald, and V. I. Norkin, “Stochastic optimization of insurance portfolios for managing exposure to catastrophic risks,” Ann. Oper. Res., Vol. 99, 207–225 (2000). DOI: https://doi.org/10.1023/A:1019244405392.
A. Ruszczynki, “Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedral,” Math. Program., Vol. 93, 195–215 (2002). DOI https://doi.org/10.1007/s10107-002-0337-7.
V. I. Norkin and S. V. Boiko, “Safety-first portfolio selection,” Cybern. Syst. Analysis, Vol. 48, No. 2, 180–191 (2012).
B. V. Norkin, “Random search of multicriterion optimum in insurance,” in: Proc. 4th Intern. Scientific Conf. of Students and Young Scientists “Theoretical and Applied Aspects of Cybernetics” (TAAC-2011. Kyiv, Nov. 24–28, 2014), Bukrek, Kyiv (2014), pp. 176–187. URL: https://www.researchgate.net/publication/301764924_Random_Search_of_Multicriterion_Optimum_in_Insurance.
B. V. Norkin, “Necessary and sufficient conditions of existence and uniqueness of solutions to integral equations of actuarial mathematics,” Cybern. Syst. Analysis, Vol. 42, No. 5, 743–749 (2006).
B. V. Norkin, “Stochastic optimal control of risk processes with Lipschitz payoff functions,” Cybern. Syst. Analysis, Vol. 50, No. 5, 774–787 (2014).
I. M. Sobol’ and R. B. Statnikov, Choosing Optimal Parameters in Multi-Criteria Problems [in Russian], Drofa, Moscow (2006).
B. V. Norkin, “Sample approximations of multiobjective stochastic optimization problems,” Optimization-online, Nov. (2014). URL: http://www.optimization-online.org/DB_HTML/2014/ll/4655.html.
B. V. Norkin, “Statistical approximation of multicriteria problems of stochastic programming,” Dopov. Nac. Akad. Nauk Ukr., No. 4, 35–41 (2015). DOI https://doi.org/10.15407/dopovidi2015.04.035.
B. V. Norkin, “On the approximation of vector optimization problems,” Kibernetika i Vych. Tekhnika, Issue 179, 35–42 (2015). URL: http://dspace.nbuv.gov.ua/bitstream/handle/123456789/86145/03-Norkin.pdf?sequence=1.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Kibernetika i Sistemnyi Analiz, No. 1, January–February, 2020, pp. 70–81.
Rights and permissions
About this article
Cite this article
Ermoliev, Y.M., Norkin, V.I. & Norkin, B.V. Stochastic Optimization Models of Actuarial Mathematics. Cybern Syst Anal 56, 58–67 (2020). https://doi.org/10.1007/s10559-020-00221-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10559-020-00221-0