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Robust Stochastic Stackelberg Differential Reinsurance and Investment Games for an Insurer and a Reinsurer with Delay

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Abstract

This paper investigates the asset-liability management problem for an ordinary robust insurance system between a reinsurer and an insurer under the Heston model with delay. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the Heston model. The insurer, as the follower of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. Under the consideration of the performance-related capital inflow/outflow, the wealth processes of the insurer and reinsurer are modeled by stochastic differential delay equations (SDDEs). This paper aims to find the equilibrium strategies for the reinsurer and insurer by maximizing the expected utility of the player’s terminal wealth with delay under the worst-case scenario of the alternative measures. By using the idea of backward induction and dynamic programming approach, the explicit expressions of the robust equilibrium strategies and value functions are derived. Finally, some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the robust optimal reinsurance and investment strategies are performed.

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References

  • A C, Li Z (2015) Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston’s SV model. Insur Math Econ 61:181–196

    Article  MathSciNet  Google Scholar 

  • A C, Lai Y, Shao Y (2018) Optimal excess-of-loss reinsurance and investment problem with delay and jump–diffusion risk process under the CEV model. J Comput Appl Math 342:317–336

    Article  MathSciNet  Google Scholar 

  • Bai Y, Zhou Z, Xiao H, Gao R, Zhong F (2019) A hybrid stochastic differential reinsurance and investment game with bounded memory. arXiv:1910.09834

  • Borch K (1960) Reciprocal reinsurance treaties. Astin Bull 1 (04):170–191

    Article  Google Scholar 

  • Branger N, Larsen LS (2013) Robust portfolio choice with uncertainty about jump and diffusion risk. J Bank Finance 37(12):5036–5047

    Article  Google Scholar 

  • Chang M, Pang T, Yang Y (2011) A stochastic portfolio optimization model with bounded memory. Math Oper Res 36(4):604–619

    Article  MathSciNet  Google Scholar 

  • Chen L, Shen Y (2018) On a new paradigm of optimal reinsurance: A stochastic Stackelberg differential game between an insurer and a reinsurer. Astin Bull 48(02):905–960

    Article  MathSciNet  Google Scholar 

  • Chen L, Shen Y (2019) Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance framework. Insur Math Econ 88:120–137

    Article  MathSciNet  Google Scholar 

  • Chen P, Yang H, Yin G (2008) Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model. Insur Math Econ 43:456–465

    Article  MathSciNet  Google Scholar 

  • Chiu MC, Li D (2006) Asset and liability management under a continuous-time mean-variance optimization framework. Insur Math Econ 39(3):330–355

    Article  MathSciNet  Google Scholar 

  • Elsanosi I, Øksendal B, Sulem A (2000) Some solvable stochastic control problems with delay. Stochast Int J Probab Stochast Process 71(1-2):69–89

    MathSciNet  MATH  Google Scholar 

  • Huang Y, Yang X, Zhou J (2017) Robust optimal investment and reinsurance problem for a general insurance company under Heston model. Math Methods Oper Res 85(2):305–326

    Article  MathSciNet  Google Scholar 

  • Keel A, Muller H (1995) Efficient portfolios in the asset liability context. Astin Bull 25(01):33–48

    Article  Google Scholar 

  • Li Z, Zeng Y, Lai Y (2012) Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s sv model. Insur Math Econ 51 (1):191–203

    Article  MathSciNet  Google Scholar 

  • Li D, Rong X, Zhao H (2016) Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model. Comput Appl Math 35(2):533–557

    Article  MathSciNet  Google Scholar 

  • Li D, Rong X, Zhao H (2017) Equilibrium excess-of-loss reinsurance–investment strategy for a mean–variance insurer under stochastic volatility model. Commun Stat-Theory Methods 46(19):9459–9475

    Article  MathSciNet  Google Scholar 

  • Li D, Shen Y, Zeng Y (2018) Dynamic derivative-based investment strategy for mean–variance asset–liability management with stochastic volatility. Insur Math Econ 78:72–86

    Article  MathSciNet  Google Scholar 

  • Maenhout PJ (2004) Robust portfolio rules and asset pricing. Rev Financ Stud 17(4):951–983

    Article  Google Scholar 

  • Pan J, Zhang Z, Zhou X (2018) Optimal dynamic mean-variance asset-liability management under the Heston model. Adv Differ Equ 2018(1):1–16

    Article  MathSciNet  Google Scholar 

  • Pan J, Hu S, Zhou X (2019) Optimal investment strategy for asset-liability management under the Heston model. Optimization 68(5):895–920

    Article  MathSciNet  Google Scholar 

  • Sharpe WF, Tint LG (1990) Liabilities-a new approach. J Portf Manag 16(2):5–10

    Article  Google Scholar 

  • Shen Y, Zeng Y (2014) Optimal investment–reinsurance with delay for mean–variance insurers A maximum principle approach. Insur Math Econ 57:1–12

    Article  MathSciNet  Google Scholar 

  • Sun Z, Guo J (2018) Optimal mean–variance investment and reinsurance problem for an insurer with stochastic volatility. Math Methods Oper Res 88(1):59–79

    Article  MathSciNet  Google Scholar 

  • Sun J, Yao H, Kang Z (2019) Robust optimal investment–reinsurance strategies for an insurer with multiple dependent risks. Insur Math Econ 89:157–170

    Article  MathSciNet  Google Scholar 

  • Wang S, Rong X, Zhao H (2016) A game between insurer and reinsurer under the Heston model. Chin J Eng Math 1:1–16

    MathSciNet  MATH  Google Scholar 

  • Xie S, Li Z, Wang S (2008) Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach. Insur Math Econ 42 (3):943–953

    Article  MathSciNet  Google Scholar 

  • Yi B, Li Z, Viens F, 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  Google Scholar 

  • Zeng Y, Li D, Chen Z, Yang Z (2018) Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility. J Econ Dyn Control 88:70–103

    Article  MathSciNet  Google Scholar 

  • Zhang Y, Wu Y, Wiwatanapataphee B, Angkola F (2020) Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. J Ind Manag Optim 16(1):71–101

    MathSciNet  MATH  Google Scholar 

  • Zhu H, Zhang C, Cao M (2020) Investment and reinsurance games between an insurer and a reinsurer under the Heston model. Chin J Manag Sci. https://doi.org/10.16381/j.cnki.issn1003-207x.2019.0270

  • Zimbidis AA (2008) Premium and reinsurance control of an ordinary insurance system with liabilities driven by a fractional Brownian motion. Scand Actuar J 2008(1):16–33

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to the anonymous referees and editors for their helpful suggestions. This research was supported by the National Natural Science Foundation of China (Nos. 71571053, 71771058), Natural Science Foundation of Guangdong Province (Nos. 2017A030313400, 2018A030313687).

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Authors and Affiliations

  1. School of Management, Guangdong Polytechnic Normal University, 510520, Guangzhou, People’s Republic of China

    Lu Yang

  2. School of Economics & Commence, Guangdong University of Technology, 510520, Guangzhou, People’s Republic of China

    Chengke Zhang & Huainian Zhu

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  1. Lu Yang
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  2. Chengke Zhang
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  3. Huainian Zhu
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Correspondence to Huainian Zhu.

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Yang, L., Zhang, C. & Zhu, H. Robust Stochastic Stackelberg Differential Reinsurance and Investment Games for an Insurer and a Reinsurer with Delay. Methodol Comput Appl Probab 24, 361–384 (2022). https://doi.org/10.1007/s11009-021-09855-9

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  • DOI: https://doi.org/10.1007/s11009-021-09855-9

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