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doi: 10.3934/jimo.2021120
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## Optimal investment-reinsurance strategy in the correlated insurance and financial markets

 School of Sciences, Hebei University of Technology, Tianjin 300401, China

* Corresponding author: Xiaoyu Xing

Received  October 2020 Revised  January 2021 Early access July 2021

Fund Project: The first author is supported by NSF grant No.12071107 and State Scholarship Fund 201906705011

Within the correlated insurance and financial markets, we consider the optimal reinsurance and asset allocation strategies. In this paper, the risk asset is assumed to follow a general continuous diffusion process driven by a Brownian motion, which correlates to the insurer's surplus process. We propose a novel approach to derive the optimal investment-reinsurance strategy and value function for an exponential utility function. To illustrate this, we show how to derive the explicit closed strategies and value functions when the risk asset is the CEV model, 3/2 model and Merton's IR model respectively.

Citation: Xiaoyu Xing, Caixia Geng. Optimal investment-reinsurance strategy in the correlated insurance and financial markets. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021120
##### References:
 [1] C. A, Y. Lai and Y. Shao, Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the CEV model, J. Comput. Appl. Math., 342 (2018), 317-336.  doi: 10.1016/j.cam.2018.03.035.  Google Scholar [2] J. Bi and J. Cai, Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets, Insurance Math. Econom., 85 (2019), 1-14.  doi: 10.1016/j.insmatheco.2018.11.007.  Google Scholar [3] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^nd$ edition, Springer-Verlag, New York, 2006.  Google Scholar [4] J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance Math. Econom., 45 (2009), 9-18.  doi: 10.1016/j.insmatheco.2009.02.006.  Google Scholar [5] A. Gu, F. G. Viens and H. Yao, Optimal robust reinsurance-investment strategies for insurers with mean reversion and mispricing, Insurance Math. Econom., 80 (2018), 93-109.  doi: 10.1016/j.insmatheco.2018.03.004.  Google Scholar [6] M. Gu, Y. Yang, S. Li and J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46 (2010), 580-587.  doi: 10.1016/j.insmatheco.2010.03.001.  Google Scholar [7] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, J. Comput. Appl. Math., 296 (2016), 443-461.  doi: 10.1016/j.cam.2015.09.032.  Google Scholar [8] B.-G. Jang and K. T. Kim, Optimal reinsurance and asset allocation under regime switching, Journal of Banking and Finance, 56 (2015), 37-47.  doi: 10.1016/j.jbankfin.2015.03.002.  Google Scholar [9] M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods for Financial Markets, 2$^nd$ edition, Springer-Verlag London, Ltd., London, 2009. doi: 10.1007/978-1-84628-737-4.  Google Scholar [10] Z. Liang, K. C. Yuen and K. C. Cheung, Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Appl. Stoch. Models Bus. Ind., 28 (2012), 585-597.  doi: 10.1002/asmb.934.  Google Scholar [11] Y. Qian and X. Lin, Run probablilities under an optimal investment and proportional reinsurance policy in a jump diffusion risk process, ANZIAM J., 51 (2009), 34-48.  doi: 10.1017/S144618110900042X.  Google Scholar [12] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 328 (2018), 414-431.  doi: 10.1016/j.cam.2017.08.001.  Google Scholar [13] G. Wells, D. Newton and D. Sanders, Milliman White Paper: Non-Life Insurance Claims in a Recession, Milliman Inc, 2009. Google Scholar [14] J. Xiao, Z. Hong and C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302-310.  doi: 10.1016/j.insmatheco.2006.04.007.  Google Scholar [15] Y. Zhang, Y. Wu, B. Wiwatanapataphee and F. Angkola, 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 (2020), 71-101.  doi: 10.3934/jimo.2018141.  Google Scholar [16] H. Zhao, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, J. Math. Anal. Appl., 437 (2016), 1036-1057.  doi: 10.1016/j.jmaa.2016.01.035.  Google Scholar [17] H. Zhao, C. Weng, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, Sci. China Math., 60 (2017), 317-344.  doi: 10.1007/s11425-015-0542-7.  Google Scholar

show all references

##### References:
 [1] C. A, Y. Lai and Y. Shao, Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the CEV model, J. Comput. Appl. Math., 342 (2018), 317-336.  doi: 10.1016/j.cam.2018.03.035.  Google Scholar [2] J. Bi and J. Cai, Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets, Insurance Math. Econom., 85 (2019), 1-14.  doi: 10.1016/j.insmatheco.2018.11.007.  Google Scholar [3] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^nd$ edition, Springer-Verlag, New York, 2006.  Google Scholar [4] J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance Math. Econom., 45 (2009), 9-18.  doi: 10.1016/j.insmatheco.2009.02.006.  Google Scholar [5] A. Gu, F. G. Viens and H. Yao, Optimal robust reinsurance-investment strategies for insurers with mean reversion and mispricing, Insurance Math. Econom., 80 (2018), 93-109.  doi: 10.1016/j.insmatheco.2018.03.004.  Google Scholar [6] M. Gu, Y. Yang, S. Li and J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46 (2010), 580-587.  doi: 10.1016/j.insmatheco.2010.03.001.  Google Scholar [7] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, J. Comput. Appl. Math., 296 (2016), 443-461.  doi: 10.1016/j.cam.2015.09.032.  Google Scholar [8] B.-G. Jang and K. T. Kim, Optimal reinsurance and asset allocation under regime switching, Journal of Banking and Finance, 56 (2015), 37-47.  doi: 10.1016/j.jbankfin.2015.03.002.  Google Scholar [9] M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods for Financial Markets, 2$^nd$ edition, Springer-Verlag London, Ltd., London, 2009. doi: 10.1007/978-1-84628-737-4.  Google Scholar [10] Z. Liang, K. C. Yuen and K. C. Cheung, Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Appl. Stoch. Models Bus. Ind., 28 (2012), 585-597.  doi: 10.1002/asmb.934.  Google Scholar [11] Y. Qian and X. Lin, Run probablilities under an optimal investment and proportional reinsurance policy in a jump diffusion risk process, ANZIAM J., 51 (2009), 34-48.  doi: 10.1017/S144618110900042X.  Google Scholar [12] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 328 (2018), 414-431.  doi: 10.1016/j.cam.2017.08.001.  Google Scholar [13] G. Wells, D. Newton and D. Sanders, Milliman White Paper: Non-Life Insurance Claims in a Recession, Milliman Inc, 2009. Google Scholar [14] J. Xiao, Z. Hong and C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302-310.  doi: 10.1016/j.insmatheco.2006.04.007.  Google Scholar [15] Y. Zhang, Y. Wu, B. Wiwatanapataphee and F. Angkola, 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 (2020), 71-101.  doi: 10.3934/jimo.2018141.  Google Scholar [16] H. Zhao, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, J. Math. Anal. Appl., 437 (2016), 1036-1057.  doi: 10.1016/j.jmaa.2016.01.035.  Google Scholar [17] H. Zhao, C. Weng, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, Sci. China Math., 60 (2017), 317-344.  doi: 10.1007/s11425-015-0542-7.  Google Scholar
Effect of $\beta$ on the optimal strategy $\pi_s^*(t)$
Effect of $\beta_0$ on the optimal strategy $\pi_s^*(t)$, when $\rho=1$
Effect of $\beta_0$ on the optimal strategy $\pi_s^*(t)$, when $\rho=-1$
Effect of $\kappa$ on the optimal strategy $\pi_s^*(t)$
Effect of $\delta$ on the optimal strategy $\pi_s^*(t)$
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