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Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

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Abstract

In this paper, we design a competition framework for two insurers with ambiguity aversion under the utility framework and investigate the resulting stochastic reinsurance game problem. Each insurer does not have perfect confidence in the drift terms of the insurance risk and chooses to purchase per-loss reinsurance to reduce her claim risk, and the reinsurance premium is determined via the mean-variance premium principle. The objective of each insurer is to find the optimal reinsurance strategy so as to maximize the ratio of expected utility of her terminal payoff to her competitor’s under the worst-case scenario. By the dynamic programming principle and corresponding Hamilton–Jacobi–Bellman–Isaacs equation, we derive the solutions for both the equilibrium reinsurance strategy and value function under the exponential utility function. In particular, we examine the existence and uniqueness of equilibrium strategy. Finally, several numerical examples are presented to illustrate the effects of competitive relationship, ambiguity aversion and some important model parameters on the equilibrium strategy, which provide useful insights for reinsurance in reality.

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Notes

  1. In some insurance optimization problems, researchers impose the condition that retained and transferred claims are non-decreasing functions of the underlying claim. However, in this paper, we can obtain the same monotonicity of \({\mathcal {H}}_i(t,Z_i)\) and \(Z_i- {\mathcal {H}}_i(t,Z_i)\) without requiring that condition a priori.

  2. \(a\wedge b\) denotes the \(\min \{a, b\}\) and \(a\vee b\) means the \(\max \{a, b\}\).

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Acknowledgements

The research of Yu Yuan was supported by the National Natural Science Foundation of China (Grant No. 12201311), the Natural Science Founding of the Jiangsu Higher Education Institutions of China (Grant No. 22KJB110021), and the Research Institute for Risk Governance and Emergency Decision-Making. The research of Caibin Zhang was supported by National Natural Science Foundation of China (Grant No. 12101299), Natural Science Foundation in Jiangsu (Grant No. BK20210668)

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Authors and Affiliations

  1. The Research Institute for Risk Governance and Emergency Decision-Making, School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, People’s Republic of China

    Yu Yuan

  2. School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, People’s Republic of China

    Kexin Wang

  3. School of Finance, Nanjing University of Finance and Economics, Nanjing, Jiangsu, People’s Republic of China

    Caibin Zhang

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  1. Yu Yuan
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Correspondence to Caibin Zhang.

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Yuan, Y., Wang, K. & Zhang, C. Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle. Ann Oper Res 335, 441–467 (2024). https://doi.org/10.1007/s10479-024-05844-6

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