论文标题
有条件反射和相关递归最佳控制问题的向后随机微分方程
Backward stochastic differential equations with conditional reflection and related recursive optimal control problems
论文作者
论文摘要
我们介绍了一种新型的反射后退随机微分方程（BSDE），该方程式对其主要解决方案组件施加了反射约束，以惯例表示为$ y $，但就其有条件的期望而言，$ \ mathbb {e} $ \ {\ MATHCAL {G} _ {T} \}。$我们因此将其称为有条件地反映BSDE的方程式（对于简短的条件RBSDE）。 Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier, and the recent developed BSDE with a mean reflection constraint, as its two special and extreme cases: they exactly correspond to $\{\mathcal{G}_{t}\}$ being the full filtration to represent complete information, and the degenerated filtration to deterministic scenario, respectively.对于有条件的RBSDE，我们通过将Snell Invelope方法与Skorokhod引理相结合，从而在轻度条件下获得了它的存在和独特性。对于线性驱动程序的情况，我们还讨论了它的连接与在存在部分信息的情况下的一系列最佳停止问题。作为副产品，获得了一种新版本的比较定理。借助这种联系，我们通过表征相关的最佳解决方案和价值来研究一类反映递归功能的最佳控制问题的弱公式。此外，在递归功能的特殊情况下，在具有侧面反射的RBSDE中，我们研究了相关随机后退递归控制和零和零和游戏的强烈表述，既在非马克维亚框架中都具有自己的利益，并且尚未得到现有文献的全面探索。
We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as $Y$ by convention, but in terms of its conditional expectation $\mathbb{E}[Y_t|\mathcal{G}_{t}]$ on a general sub-filtration $\{\mathcal{G}_{t}\}.$ We thus term such equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier, and the recent developed BSDE with a mean reflection constraint, as its two special and extreme cases: they exactly correspond to $\{\mathcal{G}_{t}\}$ being the full filtration to represent complete information, and the degenerated filtration to deterministic scenario, respectively. For conditional RBSDE, we obtain its existence and uniqueness under mild conditions by combining the Snell envelope method with Skorokhod lemma. We also discuss its connection, in the case of linear driver, to a class of optimal stopping problems in presence of partial information. As a by-product, a new version of comparison theorem is obtained. With the help of this connection, we study weak formulations of a class of optimal control problems with reflected recursive functionals by characterizing the related optimal solution and value. Moreover, in the special case of recursive functionals being RBSDE with pointwise reflections, we study the strong formulations of related stochastic backward recursive control and zero-sum games, both in non-Markovian framework, that are of their own interests and have not been fully explored by existing literatures yet.
