Stochastic differential games between two insurers with generalized mean-variance premium principle

We study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks.
Supposing that their reinsurance premium rates are calculated according to the
generalized mean-variance principle, we consider the competition between the
two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman
equations and show the existence of equilibrium strategies. For an exponential
utility maximizing game and a probability maximizing game, we obtain semiexplicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally,we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.