AI 中文总结
研究有限时域零和控制器-停止者博弈，基于非线性斯内尔包络表示及首次接触原理证明博弈存在值，给出纯概率方法得最优反馈停止规则，还应用于非零和博弈，表明零和最优规则是\(\varepsilon\)-纳什均衡组成部分。
AI 中文摘要
我们研究有限时域、零和的控制器-停止者博弈，其中停止者观察状态过程并实施反馈停止规则。基于在一篇相关论文中建立的相关博弈的非线性斯内尔包络表示，通过扩展相关的首次接触原理，我们证明了我们的博弈存在值。我们的方法是纯概率性的，为路径依赖系统产生最优反馈停止规则，同时允许基础随机微分方程存在退化。作为应用，我们考虑非零和控制器-停止者博弈，并表明零和博弈的最优反馈停止规则是每个\(\varepsilon>0\)的\(\varepsilon\)-纳什均衡的一个组成部分。
英文摘要
We investigate finite-horizon, zero-sum controller-stopper games in which the stopper observes the state process and implements a feedback stopping rule. Building on a nonlinear Snell envelope representation for a related game established in a companion paper, we prove that our game admits a value by extending the associated first-contact principle. Our approach is purely probabilistic and yields an optimal feedback stopping rule for path-dependent systems while allowing for degeneracy in the underlying stochastic differential equation (SDE). As an application, we consider nonzero-sum controller-stopper games and show that the optimal feedback stopping rule for a zero-sum game appears as a component of a $\varepsilon$-Nash equilibrium for every $\varepsilon>0$.
