Many-insurer robust games of reinsurance and investment under model uncertainty in incomplete markets

This paper studies the robust reinsurance and investment games for competitive insurers. Model uncertainty is characterized by a class of equivalent probability measures. Each insurer is concerned with relative performance under the worst-case scenario. Insurers' surplus processes are approximated by drifted Brownian motion with common and idiosyncratic insurance risks. The insurers can purchase proportional reinsurance to divide the insurance risk with the reinsurance premium calculated by the variance principle. We consider an incomplete market driven by the 4/2 stochastic volatility mode. This paper formulates the robust mean-field game for a non-linear system originating from the variance principle and the 4/2 model. For the case of an exponential utility function, we derive closed-form solutions for the $n$-insurer game and the corresponding mean-field game. We show that relative concerns lead to new hedging terms in the investment and reinsurance strategies. Model uncertainty can significantly change the insurers' hedging demands. The hedging demands in the investment-reinsurance strategies exhibit highly non-linear dependence with the insurers' competitive coefficients, risk aversion and ambiguity aversion coefficients. Finally, numerical results demonstrate the herd effect of competition.