Robust reinsurance-pricing-investment stochastic differential game between (re)insurers under the mean-variance criterion

In this article, we investigate the time-consistent robust equilibrium reinsurance-pricing-investment strategies for two competing insurers under the mean-variance criterion, who simultaneously serve as the reinsurers of each other. In order to highlight the competition, we assume that each insurer can only purchase reinsurance from her competitor and is able to set the reinsurance price at which the competitor pays. Besides, insurers invest in the financial market to manage risks and increase wealth. Concerned about the model accuracy (claims and prices of risky assets), each insurer is considered to be ambiguity-averse and tries to seek the robust equilibrium strategy maximizing the value function in the worst case. Utilizing the stochastic control theory, we establish the Hamilton-Jacobi-Bellman equations with boundary conditions. By solving them, the time-consistent robust equilibrium reinsurance-pricing-investment strategies and the robust equilibrium value functions are then derived. Subsequently, the effects of some parameters on two insurers' decision-making are analyzed economically by comparing the robust equilibrium strategies with those in two simplified cases. In addition, we discuss two kinds of losses that result from inadequate considerations. Finally, numerical examples are provided to illustrate the influences and the necessity of some parameters.