Robust reinsurance contract and investment with delay under mean-variance framework

Abstract

We consider a (non) zero-sum differential game between two insurers and determine a robust reinsurance contract from joint interests of the insurers and the reinsurer under the framework of Stackelberg differential game, that is, a hybrid stochastic differential game framework is introduced in this article. We assume that each party can invest in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a stochastic volatility model, and the reinsurer charges the premium according to both the expected value premium principle and the variance premium principle. Since the performance-related capital inflow or outflow feature is introduced, the insurers and reinsurer all aim to maximize their own time-inconsistent mean-variance criterion with delay. Using the dynamic programming approach, we derive the equilibrium investment strategy, the robust equilibrium reinsurance contract, and the corresponding equilibrium value functions under two premium principles with certain conditions, respectively. Some interesting properties of the equilibrium investment strategy are proved explicitly, and several numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium reinsurance contract.

Keywords:

• Robust reinsurance contract design
• stochastic volatility model
• mean-variance framework
• stochastic differential game
• ambiguity aversion

AMS (2010)::

• 90C39
• 91B30
• 91G80

Acknowledgments

The authors would like to thank the anonymous referees for their careful reading and helpful comments on an earlier version of this paper, which led to a considerable improvement of the presentation of the work. The authors would like to thank Zhibin Liang for helpful comments and discussions on an earlier version of the work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. This condition makes the insurers’ competing problems solvable, see, e.g., Bai et al. (2022).

Additional information

Funding

This work is supported by the National Key R&D Program of China (2022YFA1007900). The research of Xia Han is supported by the National Natural Science Foundation of China (Grant No. 12301604, 12371471) and the Fundamental Research Funds for the Central Universities, Nankai University (Grant No. 63231138). The research of Danping Li was supported by National Natural Science Foundation of China (Grant No. 12271171, 11971172, 12071147, 71931004), Shanghai Philosophy Social Science Planning Office Project (Grant No. 2022ZJB005) and Fundamental Research Funds for the Central Universities (Grant No. 2022QKT001). The research of Yu Yuan was supported by the National Natural Science Foundation of China (Grant No. 12201311), and the Natural Science Founding of the Jiangsu Higher Education Institutions of China (Grant No. 22KJB110021).