In this paper, we study some noncooperative and cooperative stochastic differential games between two insurers with reinsurance controls. The surplus processes are modeled by diffusion approximation processes and the insurers can purchase reinsurance dynamically (i.e. control the drift and diffusion terms continuously over time). We consider two types of reinsurance: quota-share (QS) reinsurance and excess-of-loss (XL) reinsurance. In the noncooperative game, one insurer tries to minimize the probability that the surplus difference of the two insurers reaches a low target before it hits a high target, while the other aims to maximize the probability. We consider two cases of the game: in the first case, both insurers purchase XL reinsurance; and in the second case, one insurer purchases QS reinsurance while the other purchases XL reinsurance. In some parameter cases, we solve the game by finding the value function and Nash equilibrium strategy explicitly. In some parameter cases, the value function and Nash equilibrium strategy do not exist and we find the sup-value and sub-value functions of the game. We also establish and solve a cooperative game in which both insurers make joint efforts to minimize the probability that the sum of surplus processes reaches a low target before it hits a high target. Numerical examples and economic implications are given to illustrate the results.

在本文中，我们研究了具有再保险控制权的两家保险公司之间的一些非合作与合作随机差异博弈。剩余过程通过扩散近似过程进行建模，保险公司可以动态购买再保险（即随时间连续控制漂移和扩散条款）。我们考虑两种类型的再保险：份额份额（QS）再保险和亏损超额（XL）再保险。在非合作博弈中，一个保险公司试图最小化两个保险公司的盈余差异在达到高目标之前就达到一个低目标的概率，而另一家则试图最大化该概率。我们考虑游戏的两种情况：第一种情况，两家保险公司都购买XL再保险；在第二种情况下，一家保险公司购买了QS再保险，而另一家则购买了XL再保险。在某些参数情况下，我们通过明确找到价值函数和纳什均衡策略来解决博弈问题。在某些参数情况下，价值函数和纳什均衡策略不存在，我们找到了博弈的高价值和次价值函数。我们还建立并解决了一种合作博弈，在该博弈中，两家保险公司共同努力，以使剩余流程之和在达到高目标之前达到低目标的可能性最小。数值例子和经济意义说明了结果。我们还建立并解决了一种合作博弈，在该博弈中，两家保险公司共同努力，以使剩余流程之和在达到高目标之前达到低目标的可能性最小。数值例子和经济意义说明了结果。我们还建立并解决了一种合作博弈，在该博弈中，两家保险公司共同努力，以使剩余流程之和在达到高目标之前达到低目标的可能性最小。数值例子和经济意义说明了结果。