Abstract
A periodic-review insurance model is considered under the following assumptions. In order to avoid ruin the insurer maintains the company surplus above a chosen level a by capital injections at the end of each period. One-period insurance claims form a sequence of independent identically distributed nonnegative random variables with finite mean. A nonproportional reinsurance is applied for minimization of total expected discounted injections during a given planning horizon of n periods. Insurance and reinsurance premiums are calculated using the expected value principle. Optimal reinsurance strategy is established. Numerical results illustrating the theoretical ones are provided for three claims distributions.
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References
Bellman R (1957) Dynamic programming. Princeton University Press, Princeton
Bulinskaya E (2003) On the cost approach in insurance. Rev Appl Ind Math 10(2):276–286. In Russian
Bulinskaya E (2012) Optimal and Asymptotically Optimal Control for Some Inventory Models. Springer Proceedings in Mathematics and Statistics, 33. In: Shiryaev, et al (eds) Prokhorov and Contemporary Probability Theory, chapter 8, pp 138–160
Bühlmann H (1970) Mathematical methods in risk theory. Springer, Berlin
Chan W, Zhang L (2006) Direct derivation of finite-time ruin probabilities in the discrete risk model with exponential or geometric claims. North Am Actuar J 10(4):269–279
Cong J, Li Z, Tan KS (2011) The optimal strategy and capital threshold of multi-period proportional reinsurance. http://www.soa.org/arch-2011-iss1-cong.pdf
Dayananda PWA (1970) Optimal reinsurance. J Appl Probab 7(1):134–156
Diasparra M, Romera R (2010) Inequalities for the ruin probability in a controlled discrete-time risk process. Eur J Oper Res 204:496–504
Dickson DCM, Waters HR (1996) Reinsurance and ruin. Insur Math Econ 19:61–80
Dickson DCM, Waters HR (2004) Some optimal dividend problems. Astin Bull 34:49–74
Dickson D, Waters H (2006) Optimal dynamic reinsurance. Austin Bull 36(2):415–432
Eisenberg J, Schmidli H (2011) Optimal control of capital injections by reinsurance with a constant rate of interest. J Appl Probab 48(3):733–748
Gerber HU (1980) An introduction to mathematical risk theory. SS Huebner Foundation
Gromov A (2013) Optimal investment strategy in the risk model with capital injections. Abstracts of the XXXI International Seminar on Stability Problems for Stochastic Models, 23-27 April, 2013, Moscow, Russia
Hipp C, Vogt M (2003) Optimal dynamic XL reinsurance. Astin Bull 33:193–208
Højgaard B, Taksar M (1998) Optimal proportional reinsurance policies for diffusion models with transaction costs. Insur Math Econ 22:41–55
Irgens C (2005) Maximizing terminal utility by controlling risk exposure: a discrete-time dynamic control approach. Scand Actuar J 2:142–160
Kaishev VK (2005) Optimal reinsurance under convex principles of premium calculation. Insur Math Econ 36:375–398
Li ZF, Cong JF (2008) Necessary conditions of the optimal multi-period proportional reinsurance strategy. J Syst Sci Math Sci 28(11):1354–1362
Prabhu NU (1998) Stochastic storage processes: queues, insurance risk, dams, and data communications, 2nd ed. Springer, New York
Schäl M (2004) On discrete-time dynamic programming in insurance: exponential utility and minimizing the ruin probability. Scand Actuar J 3:189–210
Schmidli H (2002) On minimizing the ruin probability by investment and reinsurance. Ann Appl Probab 12(3):890–907
Wei X, Hu Y (2008) Ruin probabilities for discrete time risk models with stochastic rates of interest. Stat Probab Lett 78:707–715
Yartseva DA (2009) Upper and lower bounds for dividends in the discrete model. Moscow Univ Math Bull 61(5):222–224
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The research is supported by Russian Foundation for Basic Research, grant 13-01-00653.
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Bulinskaya, E., Gusak, J. & Muromskaya, A. Discrete-time Insurance Model with Capital Injections and Reinsurance. Methodol Comput Appl Probab 17, 899–914 (2015). https://doi.org/10.1007/s11009-014-9418-3
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DOI: https://doi.org/10.1007/s11009-014-9418-3