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Asymptotic investment behaviors under a jump-diffusion risk process. (English) Zbl 1414.91164

Summary: We study an optimal investment control problem for an insurance company. The surplus process follows the Cramer-Lundberg process with perturbation of a Brownian motion. The company can invest its surplus into a risk-free asset and a Black-Scholes risky asset. The optimization objective is to minimize the probability of ruin. We show by new operators that the minimal ruin probability function is a classical solution to the corresponding HJB equation. Asymptotic behaviors of the optimal investment control policy and the minimal ruin probability function are studied for low surplus levels with a general claim size distribution. Some new asymptotic results for large surplus levels in the case with exponential claim distributions are obtained. We consider two cases of investment control: unconstrained investment and investment with a limited amount.

MSC:

91B30 Risk theory, insurance (MSC2010)
93E20 Optimal stochastic control
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[1] Azcue, P.; Muler, M., Optimal Investment Strategy to Minimize the Ruin Probability of an Insurance Company under Borrowing Constraints, Insurance: Mathematics and Economics, 44, 26-34, (2009) · Zbl 1156.91391
[2] Belkina, T.; Hipp, C.; Luo, S.; Taksar, M., Optimal Constrained Investment in the Cramer-Lundburg model, Scandinavian Actuarial Journal, 5, 383-404, (2014) · Zbl 1401.91099
[3] Belkina, T. A.; Norshteyn, M. V.; Belenky, V. Z., Analysis and Modeling of Economic Processes: The Collection of Articles, 9, Structure of Optimal Investment Strategy in a Dynamic Model for Risks with Diffusion Disturbances, 103-112, (2012), Moscow: CEMI RAS, Moscow
[4] Bellman, R., Stability Theory of Differential Equations, (2008), New York: Dover, New York
[5] Dufresne, F.; Gerber, H. U., Risk Theory for the Compound Poisson Process That Is Perturbed by Diffusion, Insurance: Mathematics and Economics, 10, 51-59, (1991) · Zbl 0723.62065
[6] Eisenberg, J., Asymptotic Optimal Investment under Interest Rate for a Class of Subexponential Distributions, Scandinavian Actuarial Journal, 8, 671-689, (2014) · Zbl 1401.91133
[7] Frolova, A.; Kabanov, Yu.; Pergamenshchikov, S., In the Insurance Business Risky Investments Are Dangerous, Finance and Stochastics, 6, 227-235, (2002) · Zbl 1002.91037
[8] Gaier, J.; Grandits, P., Ruin Probabilities in the Presence of Regularly Varying Tails and Optimal Investment, Insurance: Mathematics and Economics, 30, 211-217, (2002) · Zbl 1055.91049
[9] Gaier, J.; Grandits, P., Ruin Probabilities and Investment under Interest Force in the Presence of Regularly Varying Tails, Scandinavian Actuarial Journal, 4, 256-278, (2004) · Zbl 1091.62102
[10] Gaier, J.; Grandits, P.; Schachermayer, W., Asymptotic Ruin Probabilities and Optimal Investment, Annals of Applied Probability, 13, 1054-1076, (2003) · Zbl 1046.62113
[11] Gerber, H. U.; Yang, H., Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment, North American Actuarial Journal, 11, 159-169, (2007)
[12] Grandits, P., Minimal Ruin Probabilities and Investment under Interest Force for a Class of Subexponential Distributions, Scandinavian Actuarial Journal, 6, 401-416, (2005) · Zbl 1142.91042
[13] Hipp, C.; Plum, M., Optimal Investment for Insurers, Insurance: Mathematics and Economics, 27, 215-228, (2000) · Zbl 1007.91025
[14] Hipp, C.; Plum, M., Optimal Investment for Investors with State Dependent Income, and for Insurers, Finance and Stochastics, 7, 299-321, (2003) · Zbl 1069.91051
[15] Hipp, C.; Schmidli, H., Asymptotics of Ruin Probabilities for Controlled Risk Processes in the Small Claims Case, Scandinavian Actuarial Journal, 5, 321-335, (2004) · Zbl 1087.62116
[16] Konyukhova, N. B., Singular Cauchy Problems for Systems of Ordinary Differential Equations, U.S.S.R. Computational Mathematics and Mathematical Physics, 23, 72-82, (1983) · Zbl 0555.34002
[17] Laubis, L.; Lin, J. E., Proceedings (electronic) of 43rd Actuarial Research Conference, Optimal Investment Allocation in a Jump Diffusion Risk Model with Investment: A Numerical Analysis of Several Examples, (2008)
[18] Lin, X., Ruin Theory for Classical Risk Process That Is Perturbed by Diffusion with Risky Investments, Applied Stochastic Models in Business and Industry, 25, 33-44, (2009) · Zbl 1224.91072
[19] Luo, S., Ruin Minimization for Insurers with Borrowing Constraints, North American Actuarial Journal, 12, 143-174, (2008)
[20] Luo, S.; Taksar, M., On Absolute Ruin Minimization under a Diffusion Approximation Model, Insurance: Mathematics and Economics, 48, 123-133, (2011) · Zbl 1233.91151
[21] Lyapunov, A. M., The General Problem of the Stability of Motion, International Journal of Control, 55, 531-534, (1992)
[22] Paulsen, J.; Gjessing, H. K., Ruin Theory with Stochastic Return on Investments, Advances in Applied Probability, 29, 965-985, (1997) · Zbl 0892.90046
[23] Schmidli, H., Optimal Proportional Reinsurance Policies in a Dynamic Setting, Scandinavian Actuarial Journal, 1, 55-68, (2001) · Zbl 0971.91039
[24] Schmidli, H., On Minimizing the Ruin Probability by Investment and Reinsurance, Annals of Applied Probability, 12, 890-907, (2002) · Zbl 1021.60061
[25] Schmidli, H., On Optimal Investment and Subexponential Claims, Insurance: Mathematics and Economics, 36, 25-35, (2005) · Zbl 1110.91019
[26] Schmidli, H., Stochastic Control in Insurance, (2008), London: Springer, London · Zbl 1133.93002
[27] Segerdahl, C. O., Über Einige Risikotheoretische Fragestellungen, Scandinavian Actuarial Journal, 25, 43-83, (1942) · JFM 68.0311.02
[28] Taksar, M.; Markussen, C., Optimal Dynamic Reinsurance Policies for Large Insurance Portfolios, Finance and Stochastics, 7, 97-121, (2003) · Zbl 1066.91052
[29] Teschl, G., Ordinary Differential Equations and Dynamical Systems, (2012), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1263.34002
[30] Wasow, W., Asymptotic Expansions for Ordinary Differential Equations, (1987), New York: Dover, New York · Zbl 0169.10903
[31] Yang, H.; Zhang, L., Optimal Investment for Insurer with Jump-Diffusion Risk Process, Insurance: Mathematics and Economics, 37, 615-634, (2005) · Zbl 1129.91020
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