This paper investigates a class of non-zero-sum stochastic differential game problems between two insurance companies. The surplus process of each company is modeled by a Brownian motion where drift and volatility depend on the continuous-time Markov regime switching process. Both companies have the option of paying dividends. The objective is to maximize the expected discount utility of surplus relative to a reference point for each insurer, the gains brought by the insurer’s own dividend payout, and the losses incurred by the dividend payment of the competitor. The gains and losses are proportional to the amount of corresponding dividend payment. To find the optimal dividend policy, we relate this singular control problem to a stopping game. Further, we prove the link based on the verification theorem and show that value functions of the non-zero-sum stochastic differential problem can be derived by integrating value functions of a stopping game. Finally, we apply our results to the case with two regimes and the case without regime switching. Numerical examples are also provided.

本文研究了两家保险公司之间的一类非零和随机微分博弈问题。每个公司的盈余过程都是通过布朗运动建模的，其中漂移和波动取决于连续时间的马尔可夫政权转换过程。两家公司都可以选择支付股息。目的是使每个保险公司相对于参考点的盈余的预期折现效用最大化，保险公司自己的股息支付所带来的收益，以及竞争对手的股息支付所产生的损失。损益与相应的股利支付额成正比。为了找到最佳的分红策略，我们将此奇异控制问题与停止博弈相关联。进一步，我们基于验证定理证明了这种联系，并表明非零和随机微分问题的价值函数可以通过整合一个停止博弈的价值函数来推导。最后，我们将结果应用于具有两种制度的情况和没有制度转换的情况。还提供了数值示例。