We study a two-player nonzero-sum stochastic differential game where one player controls the state variable via additive impulses while the other player can stop the game at any time. The main goal of this work is characterize Nash equilibria through a verification theorem, which identifies a new system of quasi-variational inequalities whose solution gives equilibrium payoffs with the correspondent strategies. Moreover, we apply the verification theorem to a game with a one-dimensional state variable, evolving as a scaled Brownian motion, and with linear payoff and costs for both players. Two types of Nash equilibrium are fully characterized, i.e. semi-explicit expressions for the equilibrium strategies and associated payoffs are provided. Both equilibria are of threshold type: in one equilibrium players' intervention are not simultaneous, while in the other one the first player induces her competitor to stop the game. Finally, we provide some numerical results describing the qualitative properties of both types of equilibrium.

中文翻译：

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脉冲控制器和停止器之间的非零和随机微分博弈

我们研究了一个两人非零和随机微分博弈，其中一个玩家通过加性脉冲控制状态变量，而另一个玩家可以随时停止游戏。这项工作的主要目标是通过验证定理来表征纳什均衡，该定理确定了一个新的准变分不等式系统，其解决方案给出了对应策略的均衡收益。此外，我们将验证定理应用于具有一维状态变量的博弈，演化为缩放布朗运动，并且两个玩家的收益和成本都是线性的。完全表征了两种类型的纳什均衡，即提供了均衡策略和相关收益的半显式表达式。两个均衡都是阈值类型的：在一个均衡中，参与者的干预不是同时的，而在另一场比赛中，第一位选手诱使她的竞争对手停止比赛。最后，我们提供了一些数值结果来描述两种类型的平衡的定性特性。