We consider a general type of non-Markovian impulse control problems under adverse non-linear expectation or, more specifically, the zero-sum game problem where the adversary player decides the probability measure. We show that the upper and lower value functions satisfy a dynamic programming principle (DPP). We first prove the dynamic programming principle (DPP) for a truncated version of the upper value function in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Following this, we use an approximation based on a combination of truncation and discretization to show that the upper and lower value functions coincide, thus establishing that the game has a value and that the DPP holds for the lower value function as well. Finally, we show that the DPP admits a unique solution and give conditions under which a saddle point for the game exists. As an example, we consider a stochastic differential game (SDG) of impulse versus classical control of path-dependent stochastic differential equations (SDEs).

我们考虑不利非线性期望下的一般类型的非马尔可夫脉冲控制问题，或更具体地说，是敌方玩家决定概率度量的零和博弈问题。我们证明了上值函数和下值函数满足动态规划原理（DPP）。我们首先以简单的方式证明上值函数的截断版本的动态规划原理（DPP）。依靠统一收敛论证使我们能够展示 DPP 的一般设置。接下来，我们使用基于截断和离散化组合的近似来表明上值函数和下值函数重合，从而确定游戏具有值并且 DPP 也适用于下值函数。最后，我们表明民进党承认一个独特的解决方案，并给出了博弈存在鞍点的条件。作为一个例子，我们考虑脉冲的随机微分博弈（SDG）与路径相关的随机微分方程（SDE）的经典控制。