In the first part of this paper, we study RBSDEs in the case where the filtration is non-quasi-left-continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from general theory of processes as the Mertens decomposition of predictable strong supermartingale. In the second part, we introduce an optimal stopping problem indexed by predictable stopping times with the nonlinear predictable g expectation induced by an appropriate backward stochastic differential equation (BSDE). We establish some useful properties of ℰp,g-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.

中文翻译：

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障碍物可预测时的反射 BSDE 和非线性最优停止问题

在本文的第一部分，我们研究了过滤非准左连续且下障碍由可预测过程给出的情况下的 RBSDE。我们利用可预测环境中最优停止理论的一些结果、一般过程理论的一些工具，如可预测强超鞅的Mertens分解，证明了存在性和唯一性。在第二部分中，我们介绍了一个最优停止问题，该问题由可预测的停止时间和非线性可预测的G由适当的反向随机微分方程（BSDE）引起的期望。我们建立了一些有用的属性ℰp,G- 鞅。此外，我们展示了最优可预测停止时间的存在，并且我们根据第一部分研究的 RBSDE 的第一个分量来表征可预测值函数。