Abstract
We find optimal (from the insurer’s point of view) strategies for insurance and reinsurance in a controllable Cramér-Lundberg risk process that describes the capital dynamics of an insurance company over an extended time interval. As the optimality criterion being minimized, we use the stationary variation coefficient, taking into account additional constraints on residual risks for both insurers and reinsurer. We establish that it is best to use stop-loss reinsurance with an upper limit and insurance which is a combination of a stop-loss strategy and franchise. We derive equations that define optimal strategy parameters.
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Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., et al., Actuarial Mathematics, Itaca, Illinois: The Society of Actuaries, 1986.
Golubin, A.Yu., Matematicheskie modeli v teorii strakhovaniya: postroenie i optimizatsiya (Mathematical Models in Insurance Theory: Construction and Optimization), Moscow: ANKIL, 2003.
Hipp, C. and Vogt, M., Optimal Dynamical XL Reinsurance, ASTIN Bulletin, 2003, vol. 33, pp. 193–207.
Schmidli, H., On Minimizing the Ruin Probability by Investment and Reinsurance, Ann. Appl. Prob., 2002, vol. 12, pp. 890–907.
Belkina, T.A. and Chekanina, S.V., Optimal Investment Control in the Dynamical Insurance Model, in Sb. “Modelirovanie mekhanizmov funktsionirovaniya ekonomiki Rossii na sovremennom etape”(Proc. “Modeling the OperationMechanisms for the Russian Economy at the Modern Stage”), Moscow: Tsentr. Econom. Mat. Inst., 2001, no. 5, pp. 101–118.
Luo, Sh., Taksar, M., and Tsoi, A., On Reinsurance and Investment for Large Insurance Portfolios, Insurance Math. Econ., 2008, vol. 42, pp. 434–444.
Gollier, C., Insurance and Precautionary Capital Accumulation in a Continuous-Time Model, J. Risk Insurance, 1994, vol. 61, pp. 78–95.
Golubin, A.Y., Optimal Insurance and Reinsurance Policies in the Risk Process, ASTIN Bulletin, 2008, vol. 38, no. 2, pp. 383–398.
Golubin, A.Yu. and Gridin, V.N., Optimal Insurance Strategies in a Risk Process with Restrictions on Policyholder Risks, Autom. Remote Control, 2010, vol. 71, no. 8, pp. 1578–1589.
Golubin, A.Yu., Gridin, V.N., and Gazov, A.I., Optimization of Risk Bearing in a Statistical Model with Reinsurance, Autom. Remote Control, 2009, vol. 70, no. 8, pp. 1395–1404.
Fleming, W.H. and Rishel, R.W., Deterministic and Stochastic Optimal Control, New York: Springer-Verlag, 1975. Translated under the title Optimal’noe upravlenie determinirovannymi i stokhasticheskimi sistemami, Moscow: Mir, 1978.
Lehmann, E.L., Testing Statistical Hypotheses, New York: Wiley, 1959. Translated under the title Proverka statisticheskikh gipotez, Moscow: Nauka, 1964.
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Original Russian Text © A.Yu. Golubin, V.N. Gridin, 2012, published in Avtomatika i Telemekhanika, 2012, No. 9, pp. 111–123.
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Golubin, A.Y., Gridin, V.N. Optimizing insurance and reinsurance in the dynamic Cramér-Lundberg model. Autom Remote Control 73, 1529–1538 (2012). https://doi.org/10.1134/S000511791209007X
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DOI: https://doi.org/10.1134/S000511791209007X