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

Wuyuan Jiang; Zechao Miao; Jun Liu; Wuyuan Jiang; Zechao Miao; Jun Liu

doi:10.3934/math.20241672

AIMS Mathematics

2024, Volume 9, Issue 12: 35181-35217. doi: 10.3934/math.20241672

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Research article

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

Department of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

Received: 10 October 2024 Revised: 11 November 2024 Accepted: 26 November 2024 Published: 16 December 2024
MSC : 91B30, 93E20

In this paper, we consider the problem of optimal investment-reinsurance for the insurer and reinsurer under the stochastic volatility model. The surplus process of the insurer is described by a diffusion model. The insurer can purchase proportional reinsurance from the reinsurer and the premium charged by the insurer and reinsurer follows the variance principle. Both the insurer and reinsurer are allowed to invest in risk-free assets and risky assets, and the market price of risk depends on a Markovian, affine-form, and square-root stochastic factor process. Our goal is to maximize the joint exponential utility of the terminal wealth of the insurer, and reinsurer over a certain period of time. By solving the HJB equation, we obtain the optimal investment-reinsurance strategies, and present the proof of the verification theorem. Finally, we demonstrate a numerical analysis, and the economic implications of our findings are illustrated.
- reinsurance and investment,
- variance premium principle,
- stochastic volatility,
- joint exponential utility
Citation: Wuyuan Jiang, Zechao Miao, Jun Liu. Optimal investment and reinsurance for the insurer and reinsurer with the joint exponential utility[J]. AIMS Mathematics, 2024, 9(12): 35181-35217. doi: 10.3934/math.20241672

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Abstract

In this paper, we consider the problem of optimal investment-reinsurance for the insurer and reinsurer under the stochastic volatility model. The surplus process of the insurer is described by a diffusion model. The insurer can purchase proportional reinsurance from the reinsurer and the premium charged by the insurer and reinsurer follows the variance principle. Both the insurer and reinsurer are allowed to invest in risk-free assets and risky assets, and the market price of risk depends on a Markovian, affine-form, and square-root stochastic factor process. Our goal is to maximize the joint exponential utility of the terminal wealth of the insurer, and reinsurer over a certain period of time. By solving the HJB equation, we obtain the optimal investment-reinsurance strategies, and present the proof of the verification theorem. Finally, we demonstrate a numerical analysis, and the economic implications of our findings are illustrated.

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