Non-zero-sum reinsurance and investment game between two mean-variance insurers under the CEV model

Abstract

This paper considers a non-zero-sum stochastic differential game between two competitive mean-variance insurers, who aim to seek the time-consistent reinsurance and investment strategies. These two insurers are allowed to purchase proportional reinsurance to mitigate individual claim risks; and can invest in one risk-free asset and one risky asset whose price dynamics follows the constant elasticity of variance model. The main objective of each insurer is to maximizing the mean-variance utility of his relative terminal wealth with respect to that of his competitor. Applying the techniques of stochastic control theory, we derive the Nash equilibrium reinsurance and investment strategies explicitly and present the corresponding verification theorem. Furthermore, Nash equilibrium strategies and value functions are also provided under the diffusion approximation model. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies and draw some economic interpretations from these results.

Keywords:

• Non-zero-sum stochastic differential game
• relative performance
• CEV model
• Nash equilibrium
• extended Hamilton–Jacobi–Bellman equation

Acknowledgments

The authors are very grateful to the anonymous referees and editors for their helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [71571053, 71673061, 71940012]; The Ministry of education of Humanities and Social Science project [18YJC790003]; Natural Science Foundation of Guangdong Province [2018A030313687].