Abstract
In this work, we study the optimal risk sharing problem for an insurer between himself and a reinsurer in a dynamical insurance model known as the Kramer–Lundberg risk process, which, unlike known models, models not per claim reinsurance but rather periodic reinsurance of damages over a given time interval. Here we take into account a natural upper bound on the risk taken by the reinsurer. We solve optimal control problems on an infinite time interval for mean-variance optimality criteria: a linear utility functional and a stationary variation coefficient. We show that optimal reinsurance belongs to the class of total risk reinsurances. We establish that the most profitable reinsurance is the stop-loss reinsurance with an upper limit. We find equations for the values of parameters in optimal reinsurance strategies.
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References
Bowers, N.L., Gerber, H.U., Hickman, J.C., et al., Actuarial Mathematics, Itaca, Illinois: The Society of Actuaries, 1986.
Hipp, C. and Vogt, M., Optimal Dynamical XL Reinsurance, ASTIN Bull., 2003, vol. 33, pp. 193–207.
Hipp, C. and Taksar, M., Optimal Non-Proportional Reinsurance Control, Insurance Math. Econom., 2010, vol. 47, pp. 246–254.
Belkina, T.A. and Chekanina, S.V., Optimal Control of Investments in a Dynamical Insurance Model, in Modeling the Mechanisms of the Modern Russian Economy, Moscow: TsEMI RAN, 2001, no. 5, pp. 101–118.
Luo, Sh., Taksar, M., and Tsoi, A., On Reinsurance and Investment for Large Insurance Portfolios, Insurance Math. Econom., 2008, vol. 42, pp. 434–444.
Golubin, A.Y., Optimal Insurance and Reinsurance Policies in the Risk Process, ASTIN Bull., 2008, vol. 38, no. 2, pp. 383–398.
Golubin, A.Y. and Gridin, V.N., Optimizing Insurance and Reinsurance in the Dynamic Kramer- Lundberg Model, Autom. Remote Control, 2012, vol. 73, no. 9, pp. 1529–1538.
Guerra, M. and Centeno, M.L., Optimal Reinsurance Policy: The Adjustment Coefficient and the Expected Utility Criteria, Insurance Math. Econom., 2008, vol. 42, pp. 529–539.
Li, Y. and Li, Z. Optimal Time-consistent Investment and Reinsurance Strategies for Mean variance Insurers with State Dependent Risk Aversion, Insurance Math. Econom., 2013, vol. 53, pp. 86–97.
Mak, T., Matematika riskovogo strakhovaniya (Mathematics of Risk Insurance), Moscow: ZAO “Olimp-Biznes,” 2005.
Fleming, W.H. and Rishel, R.W., Deterministic and Stochastic Optimal Control, Berlin: Springer-Verlag, 1975. Translated under the title Optimal’noe upravlenie determinirovannymi i stokhasticheskimi sistemami, Moscow: Mir, 1978.
Markowitz, N., Mean-Variance Analysis in Portfolio Choice and Capital Markets, Cambridge: Blackwell, 1990.
Gut, A., Probability: A Graduate Course, New York: Springer, 2005.
Tucker, H.G., A Graduate Course in Probability, New York: Academic, 1967.
Leman, E., Proverka statisticheskikh gipotez (Testing Statistical Hypotheses), Moscow: Nauka, 1964.
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Original Russian Text © A.Y. Golubin, 2017, published in Avtomatika i Telemekhanika, 2017, No. 7, pp. 110–124.
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Golubin, A.Y. Risk process with a periodic reinsurance: Choosing an optimal reinsurance strategy of a total risk. Autom Remote Control 78, 1264–1275 (2017). https://doi.org/10.1134/S0005117917070086
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DOI: https://doi.org/10.1134/S0005117917070086