论文标题
非线性期望下的非马尔可脉冲控制
Non-Markovian Impulse Control Under Nonlinear Expectation
论文作者
论文摘要
我们考虑在不利的非线性期望下的一般类型的非马克维亚冲动控制问题，或者更具体地说，是对手玩家决定概率度量的零和零游戏问题。我们表明，上下值函数满足动态编程原理（DPP）。我们首先以直接的方式证明了上值函数的截断版本的动态编程原理（DPP）。依靠统一的收敛参数，使我们能够为一般设置显示DPP。之后，我们使用基于截断和离散化的组合的近似值来表明上下值的函数重合，从而确定游戏具有值，并且DPP也具有下层值函数。最后，我们表明DPP承认了一个独特的解决方案，并提供了在游戏中存在鞍点的条件。 举例来说，我们考虑了脉冲的随机差分游戏（SDG）与路径依赖性随机微分方程（SDE）的经典控制。
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).
