Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility under the CEV model

Ling Chen; Xiang Hu; Mi Chen; Ling Chen; Xiang Hu; Mi Chen

doi:10.3934/math.2023786

AIMS Mathematics

2023, Volume 8, Issue 7: 15383-15410. doi: 10.3934/math.2023786

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Research article

Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility under the CEV model

Ling Chen ¹,
Xiang Hu ²,
Mi Chen ^{1
,
,}

1.
School of Mathematics and Statistics & FJKLMAA, Fujian Normal University, Fuzhou 350117, China
2.
School of Finance, Zhongnan University of Economics and Law, Nanhu Road, Wuhan 430073, China

Received: 22 March 2023 Revised: 16 April 2023 Accepted: 20 April 2023 Published: 26 April 2023
MSC : 91B05, 91G05

This paper considers the problem of optimal investment-reinsurance for the insurer and reinsurer under the constant elasticity of variance (CEV) model. It is assumed that the net claims process is approximated by a diffusion process, both the insurer and reinsurer can invest in risk-free assets and risky assets. We use the variance premium principle to calculate the premiums of the insurer and reinsurer, and the reinsurance proportion is constrained by the net profit condition. Our objective is to maximize the joint exponential utility of the insurer and reinsurer's terminal wealth for a fixed time. By solving the HJB equation, we obtain the explicit expressions of the optimal investment-reinsurance strategy and value function. We find that the optimal reinsurance strategy can be divided into many cases and is related to the risk aversion coefficient of the insurer and reinsurer, but independent of the price of risky assets. Furthermore, we give the proof of the verification theorem. Finally, we demonstrate a numerical analysis to explain the results.
- investment,
- reinsurance,
- variance premium principle,
- net profit condition,
- CEV model,
- joint exponential utility
Citation: Ling Chen, Xiang Hu, Mi Chen. Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility under the CEV model[J]. AIMS Mathematics, 2023, 8(7): 15383-15410. doi: 10.3934/math.2023786

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Abstract

This paper considers the problem of optimal investment-reinsurance for the insurer and reinsurer under the constant elasticity of variance (CEV) model. It is assumed that the net claims process is approximated by a diffusion process, both the insurer and reinsurer can invest in risk-free assets and risky assets. We use the variance premium principle to calculate the premiums of the insurer and reinsurer, and the reinsurance proportion is constrained by the net profit condition. Our objective is to maximize the joint exponential utility of the insurer and reinsurer's terminal wealth for a fixed time. By solving the HJB equation, we obtain the explicit expressions of the optimal investment-reinsurance strategy and value function. We find that the optimal reinsurance strategy can be divided into many cases and is related to the risk aversion coefficient of the insurer and reinsurer, but independent of the price of risky assets. Furthermore, we give the proof of the verification theorem. Finally, we demonstrate a numerical analysis to explain the results.

References

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