Time Consistent Mean-Variance Portfolio Games

In this paper we consider portfolio selection games between two investors with different risk aversion parameters under different financial markets.Firstly, we study the portfolio selection problem between two investors in a simplified financial market. The financial market consists of a risk-free asset and two correlative risky assets. The time consistent mean-variance stochastic differential portfolio problems between two investors are considered under geometric Brown motion model and Heston stochastic volatility model, respectively. By using the dynamic programming approach (HJB equations), the verification theorems are obtained, and the optimal investment strategies and optimal value functions of the two investors in the two models are expressed in explicit forms, respectively. Then a time consistent mean-variance proportional reinsurance games are investigated in the simplified markets consisting of two insurance companies, and a risk free asset is considered only. In order to mitigate risk, the insurance companies choose to purchase a proportional reinsurance. By using the dynamic programming approach (HJB equations), the verification theorems are obtained, and the optimal proportional reinsurance strategies and optimal value functions of the two insurance companies are obtained in closed-forms. Finally, some numerical simulations are provided to display the relationships between the optimal strategies and the main parameters of the model.