Time-Consistent Investment and Reinsurance Strategies for Mean–Variance Insurers in N-Agent and Mean-Field Games

In this study, we investigate the competition among insurers under the mean–variance criterion. The optimization problems are formulated for finite and infinite insurers. The surplus processes of the insurers are characterized by jump-diffusion processes with common and idiosyncratic insurance risks. The insurers can purchase a reinsurance business to divide the insurance risk. In the financial market, the insurers decide the proportion of their fund to be retained as cash and to be invested in a stock characterized by a jump-diffusion process with common and idiosyncratic financial risks. The insurers compete with each other and are concerned with the relative performance. By the extended Hamilton-Jacobi-Bellman equation, the explicit forms of the n-agent equilibrium and mean-field equilibrium (MFE) are obtained in the n-agent case and mean-field case, respectively. Our results show that the MFE of the reinsurance strategy is composed of two parts, one part associated with the individual preference and the other associated with the common insurance shock. Meanwhile, the MFE of the investment strategy is composed of three parts: the individual preference, common market risks, and common shocks. Numerical examples are presented at the end of this article to demonstrate the effects of different parameters on the MFE. The results reveal that the insurers become convergent in a competitive environment.