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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 and Management Optimization, doi: 10.3934/jimo.2021120
References:
[1]

C. AY. 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.

[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.

[3]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^nd$ edition, Springer-Verlag, New York, 2006.

[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.

[5]

A. GuF. 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.

[6]

M. GuY. YangS. 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.

[7]

Y. HuangX. 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.

[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.

[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.

[10]

Z. LiangK. 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.

[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.

[12]

Y. WangX. 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.

[13]

G. Wells, D. Newton and D. Sanders, Milliman White Paper: Non-Life Insurance Claims in a Recession, Milliman Inc, 2009.

[14]

J. XiaoZ. 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.

[15]

Y. ZhangY. WuB. 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.

[16]

H. ZhaoY. 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.

[17]

H. ZhaoC. WengY. 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.

show all references

References:
[1]

C. AY. 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.

[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.

[3]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^nd$ edition, Springer-Verlag, New York, 2006.

[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.

[5]

A. GuF. 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.

[6]

M. GuY. YangS. 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.

[7]

Y. HuangX. 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.

[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.

[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.

[10]

Z. LiangK. 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.

[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.

[12]

Y. WangX. 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.

[13]

G. Wells, D. Newton and D. Sanders, Milliman White Paper: Non-Life Insurance Claims in a Recession, Milliman Inc, 2009.

[14]

J. XiaoZ. 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.

[15]

Y. ZhangY. WuB. 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.

[16]

H. ZhaoY. 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.

[17]

H. ZhaoC. WengY. 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.

Figure 1.  Effect of $ \beta $ on the optimal strategy $ \pi_s^*(t) $
Figure 2.  Effect of $ \beta_0 $ on the optimal strategy $ \pi_s^*(t) $, when $ \rho=1 $
Figure 3.  Effect of $ \beta_0 $ on the optimal strategy $ \pi_s^*(t) $, when $ \rho=-1 $
Figure 4.  Effect of $ \kappa $ on the optimal strategy $ \pi_s^*(t) $
Figure 5.  Effect of $ \delta $ on the optimal strategy $ \pi_s^*(t) $
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