Abstract
We consider the optimal choice problem by a risk-bearing function for an insurer to divide risks between him and his clients in a dynamic insurance model, the so-called Cramer-Lundberg risk process. In this setting, we take into account restrictions imposed on policyholder risks, either on the mean value or a constraint with probability one. We solve the optimal control problem on an infinite time interval for the optimality criterion of the stationary coefficient of variation. We show that in the model with a restriction on average risk the stop-loss insurance strategy will be most profitable. For a probability one restriction, the optimal insurance is a combination of a stop-loss strategy and a deductible. We show that these results extend to a number of problems with other optimality criteria, e.g., the problems of maximizing unit utility and minimizing the probability of deviating from the mean value.
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References
Bowers, N.L., Gerber, H.U., Hickman, J.C., et al., Actuarial Mathematics, Itaca: 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.
Arrow, K.J., Essays in the Theory of Risk Bearing, Chicago: Markham, 1971.
Raviv, A., The Design of an Optimal Insurance Policy, Am. Econom. Rev., 1979, pp. 84–96.
Golubin, A.Yu., Gridin, V.N., and Gazov, A.I., Optimization of Risk Bearing in a Statistical Model with Reinsurance, Autom. Remote Control, 2009, no. 8, pp. 1385–1395.
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. Probab., 2002, vol. 12, pp. 890–907.
Belkina, T.A. and Chekanina, S.P., Optimal Investment Management in a Dynamic Insurance Model, in Modelirovanie mekhanizmov funktsionirovaniya ekonomiki Rossii na sovremennom etape (Modeling the Mechanisms of Russian Economy Operation on the Present Stage), Moscow: Tsentr. Ekonom.-matemat. Inst., 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.
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.
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.
Markowitz, H., Mean-Variance Analysis in Portfolio Choice and Capital Markets, Cambridge: Blackwell, 1990.
Shorgin, S.Ya., On the Accuracy of the Normal Approximation of Distributions of Random Sums with Infinitely Divisible Indices, Teor. Veroyatn. Primen., 1996, vol. 41, no. 4, pp. 920–926.
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Original Russian Text © A.Yu. Golubin, V.N. Gridin, 2010, published in Avtomatika i Telemekhanika, 2010, No. 8, pp. 79–91.
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Golubin, A.Y., Gridin, V.N. Optimal insurance strategies in a risk process with restrictions on policyholder risks. Autom Remote Control 71, 1578–1589 (2010). https://doi.org/10.1134/S0005117910080072
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DOI: https://doi.org/10.1134/S0005117910080072