Optimal Reinsurance and Investment Problems for Two Cooperative Insurers and One Reinsurer

Xinya He; Shumin Chen; Ailing Gu; Hui Chen

doi:10.1007/s11518-025-5697-1

Journal of Systems Science and Systems Engineering ›› :1 -25. DOI: 10.1007/s11518-025-5697-1

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Optimal Reinsurance and Investment Problems for Two Cooperative Insurers and One Reinsurer

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¹ School of Management, Guangdong University of Technology, 510520, Guangzhou, China

² School of Mathematics and Statistics, Guangdong University of Technology, 510520, Guangzhou, China

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^a 771090379@qq.com

Xinya He is a doctoral student at the School of Management, Guangdong University of Technology. She holds a master of science degree from the School of Mathematics and Statistics, Guangdong University of Technology, and a bachelor of economics degree from the School of Statistics and Mathematics, Yunnan University of Finance and Economics. Her research interests include financial engineering and insurance risk management.

Shumin Chen is a professor and doctoral supervisor at the School of Management, Guangdong University of Technology. He has published over twenty academic papers in renowned journals such as Journal of Banking & Finance, SIAM Journal on Financial Mathematics, etc. He has presided over three National Natural Science Foundation of China projects (including youth and general programs), two Guangdong Provincial Natural Science Foundation projects, and several other significant research projects. Professor Chen has long been dedicated to research in fields such as financial engineering and actuarial science.

Ailing Gu is an associate professor at the School of Mathematics and Statistics, Guangdong University of Technology. Her research focuses on insurance companies’ optimal investment, including optimal reinsurance and investment strategies under general and dependent risk models.

Hui Chen is a master’s student at the School of Management, Guangdong University of Technology. She holds a bachelor of management degree from the School of Mathematics and Computer Science (Software College), South China Agricultural University. Her research interests include financial engineering, insurance, and game theory.

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Abstract

The purpose of this paper is to investigate an optimal excess-of-loss reinsurance contract and investment problem involving two mutually cooperative insurers and a reinsurer, within the framework of the Stackelberg game. In the reinsurance contract, the two insurers are permitted to purchase excess-of-loss reinsurance from the reinsurer, who sets the pricing level for the reinsurance. Assume that the two insurers and the reinsurer invest their surpluses in a financial market comprising a risk-free asset and a risky asset, whose price process is described by geometric Brownian motion. Under the criterion of maximizing the expected exponential utility of their terminalwealth, the explicit expressions for the optimal strategies and the corresponding value functions are derived using techniques from optimal control theory and the dynamic programming method. Moreover, to further enhance the research system of reinsurance contract problems within the Stackelberg game framework, we also consider the optimization problem under the proportional reinsurance model. Finally,we present theoretical analyses and numerical examples to illustrate the economic intuition and insights gained from our results. An interesting finding is that the forms of the optimal proportional reinsurance strategy and the optimal excess-of-loss reinsurance strategy are remarkably similar, with differences primarily stemming from the type of reinsurance chosen by the insurers. In addition, we also discover that the two cooperating insurers will develop their own optimal reinsurance strategies based on their respective importance within the cooperative group.

Electronic Supplementary Material

Supplementary material is available in the online version of this article at https://doi.org/10.1007/s11518-025-5697-1 and is accessible for authorized users.

Keywords

Stackelberg game / expected exponential utility / optimal control / dynamic programming / optimal reinsurance contract / investment strategy

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Xinya He, Shumin Chen, Ailing Gu, Hui Chen. Optimal Reinsurance and Investment Problems for Two Cooperative Insurers and One Reinsurer. Journal of Systems Science and Systems Engineering 1-25 DOI:10.1007/s11518-025-5697-1

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