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Robust optimal investment and reinsurance problem for a general insurance company under Heston model

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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

    Article  Google Scholar 

  • Bai LH, Guo JY (2008) Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insur Math Econ 42:968–975

    Article  MathSciNet  MATH  Google Scholar 

  • Borch K (1960) Reciprocal reinsurance treaties. ASTIN Bull 1:171–191

    Article  Google Scholar 

  • Borch K (1969) The optimal reinsurance treaties. ASTIN Bull 5:293–297

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Chacko G, Viceira LM (2005) Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets. Rev Financ Stud 8:1369–1402

    Article  Google Scholar 

  • Chen SM, Li ZF, Li KM (2010) Optimal investment-reinsurance for an insurance company with VaR constraint. Insur Math Econ 47:144–153

    Article  MathSciNet  MATH  Google Scholar 

  • Dimitrova DS, Kaishev VK (2010) Optimal joint survival reinsurance: an efficient frontier approach. Insur Math Econ 47:27–35

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Gerber HU, Shiu ESW (2006) On optimal dividends: from reflection to refraction. J Comput Appl Math 186:4–22

    Article  MathSciNet  MATH  Google Scholar 

  • Grandell J (1991) Aspects of risk theory. Springer, New York

    Book  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Harrison MJ (1977) Ruin problems with compounding assets. Stoch Process Appl 5:67–79

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Iglehart DG (1969) Diffusion approximations in collective risk theory. J Appl Probab 6:285–292

    Article  MathSciNet  MATH  Google Scholar 

  • Kaishev VK, Dimitrova DS (2006) Excess of loss reinsurance under joint survival optimality. Insur Math Econ 39:376–389

    Article  MathSciNet  MATH  Google Scholar 

  • Kraft H (2005) Optimal portfolios and Heston’s stochastic volatility model: an explicit solution for power utility. Quant Finance 5:303–313

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Liang ZB, Bai LH, Guo JY (2011) Optimal investment and proportional reinsurance with constrained control variables. Optim Control Appl Methods 32:587–608

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Liu J (2007) Portfolio selection in stochastic environment. Rev Financ Stud 20:1–39

    Article  Google Scholar 

  • Liu J, Pan J (2003) Dynamic derivative strategies. J Financ Econ 69:401–430

    Article  Google Scholar 

  • Liu HN (2010) Robust consumption and portfolio choice for time varying investment opportunities. Ann Finance 6:435–454

    Article  MATH  Google Scholar 

  • Maenhout PJ (2004) Robust portfolio rules and asset pricing. Rev Financ Stud 17:951–983

    Article  Google Scholar 

  • Maenhout PJ (2006) Robust portfolio rules and dectection-error probabilities for a mean-reverting risk premium. J Econ Theory 128:136–163

    Article  MathSciNet  MATH  Google Scholar 

  • Mataramvura S, Øksendal B (2008) Risk minimizing and HJBI equations for stochastic differential games. Stoch Int J Probab Stoch Process 80:317–337

    Article  MathSciNet  MATH  Google Scholar 

  • Promislow DS, Young VR (2005) Minimizing the probability of ruin when claims follow Brownian motion with drift. N Am Actuar J 9:109–128

    Article  MathSciNet  MATH  Google Scholar 

  • Uppal R, Wang T (2003) Model misspecification and underdiversification. J Finance 58:2465–2486

    Article  Google Scholar 

  • Yang HL, Zhang LH (2005) Optimal Investment for Insurer with jump-diffusion risk process. Insur Math Econ 37:615–634

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Zeng XD, Taksar M (2013) A stochastic volatility model and optimal portfolio selection. Quant Finance 13:1547–1558

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang X, Siu TK (2009) Optimal investment and reinsurance of an insurer with model uncertainty. Insur Math Econ 45:81–88

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

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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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Authors and Affiliations

  1. School of Business, Hunan Normal University, Changsha, 410081, China

    Ya Huang

  2. Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, China

    Xiangqun Yang & Jieming Zhou

Authors
  1. Ya Huang
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  2. Xiangqun Yang
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  3. Jieming Zhou
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Correspondence to Jieming Zhou.

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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

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