Optimal risk sharing with correlated insurance businesses in a Stackelberg-Nash differential game

In this paper, we investigate the optimal risk sharing problem for two insurers under the framework of Stackelberg-Nash differential game. More specifically, two insurers transfer their businesses to each other for achieving the goal of win-win, where both of them act as the leader for pricing while the follower for choosing their own retention level. Based on the game theoretic equilibrium setting and dynamic programming principle, the explicit optimal strategies are derived. We find that insurers will cooperate more eagerly when there is a stronger negative correlation between the businesses of both parties. In order to explore the advantages of risk sharing, we also investigate the optimal reinsurance problem in a traditional Stackelberg game framework. Risk sharing is found to be more advantageous than reinsurance in many cases, especially when the businesses have significant differences, such as a strong negative correlation or a large/small volatility ratio, which means that one of the two businesses is relatively stable while the other fluctuates greatly. Further analysis is given to show the effects of model parameters and the economics interpretations behind them. It is interesting to find that risk-aversion coefficient plays a key role in this Stackelberg-Nash differential game, and the conclusions confirm an obvious fact, that is, risk-averse individuals tend to be more hesitant and conservative when making a decision.