Abstract
In this paper, we study a robust optimal investment and reinsurance problem for a general insurance company which contains an insurer and a reinsurer. Assume that the claim process described by a Brownian motion with drift, the insurer can purchase proportional reinsurance from the reinsurer. Both the insurer and the reinsurer can invest in a financial market consisting of one risk-free asset and one risky asset whose price process is described by the Heston model. Besides, the general insurance company’s manager will search for a robust optimal investment and reinsurance strategy, since the general insurance company faces model uncertainty and its manager is ambiguity-averse in our assumption. The optimal decision is to maximize the minimal expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s surplus processes. By using techniques of stochastic control theory, we give sufficient conditions under which the closed-form expressions for the robust optimal investment and reinsurance strategies and the corresponding value function are obtained.
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References
Anderson EL, Hansen LP, Sargent TJ (1999) Robustness detection and the price of risk. Working Paper, University of Chicago. https://files.nyu.edu/ts43/public/research/.svn/text-base/ahs3.svn-base
Anderson EL, Hansen LP, Sargent TJ (2003) A quartet of semi-groups for model specification, robustness, prices of risk, and model detection. J Eur Econ Assoc 1:68–123
Bai LH, Guo JY (2008) Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insur Math Econ 42:968–975
Borch K (1960) Reciprocal reinsurance treaties. ASTIN Bull 1:171–191
Borch K (1969) The optimal reinsurance treaties. ASTIN Bull 5:293–297
Browne S (1995) Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Math Oper Res 20:937–957
Cai J, Fang Y, Li Z, Willmot GE (2013) Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability. J Risk Insur 80:145–168
Chacko G, Viceira LM (2005) Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets. Rev Financ Stud 8:1369–1402
Chen SM, Li ZF, Li KM (2010) Optimal investment-reinsurance for an insurance company with VaR constraint. Insur Math Econ 47:144–153
Dimitrova DS, Kaishev VK (2010) Optimal joint survival reinsurance: an efficient frontier approach. Insur Math Econ 47:27–35
Fang Y, Qu Z (2014) Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability. IMA J Manag Math 25:89–103
Gerber HU, Shiu ESW (2006) On optimal dividends: from reflection to refraction. J Comput Appl Math 186:4–22
Grandell J (1991) Aspects of risk theory. Springer, New York
Gu AL, Guo XP, Li ZF, Zeng Y (2012) Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model. Insur Math Econ 51:674–684
Harrison MJ (1977) Ruin problems with compounding assets. Stoch Process Appl 5:67–79
Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6:327–343
Iglehart DG (1969) Diffusion approximations in collective risk theory. J Appl Probab 6:285–292
Kaishev VK, Dimitrova DS (2006) Excess of loss reinsurance under joint survival optimality. Insur Math Econ 39:376–389
Kraft H (2005) Optimal portfolios and Heston’s stochastic volatility model: an explicit solution for power utility. Quant Finance 5:303–313
Li D, Rong X, Zhao H (2014a) The optimal investment problem for an insurer and a reinsurer under the constant elasticity of variance model. IMA J Manag Math 2014:1–26
Li D, Rong X, Zhao H (2014b) Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model. Comput Appl Math 2014:1–25
Li Z, Zeng Y, Lai Y (2012) Optimal time-consistent investment and reinsurance strategies for insurers under Hestons SV model. Insur Math Econ 51:191–203
Liang ZB, Bai LH, Guo JY (2011) Optimal investment and proportional reinsurance with constrained control variables. Optim Control Appl Methods 32:587–608
Lin X, Zhang CH, Siu TK (2012) Stochastic differential portfolio games for an insurer in a jump-diffusion risk process. Math Methods Oper Res 75:83–100
Liu J (2007) Portfolio selection in stochastic environment. Rev Financ Stud 20:1–39
Liu J, Pan J (2003) Dynamic derivative strategies. J Financ Econ 69:401–430
Liu HN (2010) Robust consumption and portfolio choice for time varying investment opportunities. Ann Finance 6:435–454
Maenhout PJ (2004) Robust portfolio rules and asset pricing. Rev Financ Stud 17:951–983
Maenhout PJ (2006) Robust portfolio rules and dectection-error probabilities for a mean-reverting risk premium. J Econ Theory 128:136–163
Mataramvura S, Øksendal B (2008) Risk minimizing and HJBI equations for stochastic differential games. Stoch Int J Probab Stoch Process 80:317–337
Promislow DS, Young VR (2005) Minimizing the probability of ruin when claims follow Brownian motion with drift. N Am Actuar J 9:109–128
Uppal R, Wang T (2003) Model misspecification and underdiversification. J Finance 58:2465–2486
Yang HL, Zhang LH (2005) Optimal Investment for Insurer with jump-diffusion risk process. Insur Math Econ 37:615–634
Yi B, Li ZF, Viens FG, Zeng Y (2013) Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model. Insur Math Econ 53:601–614
Zeng XD, Taksar M (2013) A stochastic volatility model and optimal portfolio selection. Quant Finance 13:1547–1558
Zhang X, Siu TK (2009) Optimal investment and reinsurance of an insurer with model uncertainty. Insur Math Econ 45:81–88
Zhao H, Rong XM, Zhao YG (2013) Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model. Insur Math Econ 53:504–514
Acknowledgements
This work is supported by a grant from the National Natural Science Foundation of China (grant Nos. 11301303, 11571189, 11601147, 11671132); and the Scientific Research Fund of Hunan Provincial Education Department, China (grant No. 16C0953).
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Huang, Y., Yang, X. & Zhou, J. Robust optimal investment and reinsurance problem for a general insurance company under Heston model. Math Meth Oper Res 85, 305–326 (2017). https://doi.org/10.1007/s00186-017-0570-8
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DOI: https://doi.org/10.1007/s00186-017-0570-8