The optimal stopping problem is one of the core problems in financial markets, with broad applications such as pricing American and Bermudan options. The deep BSDE method [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] has shown great power in solving high-dimensional forward-backward stochastic differential equations (FBSDEs), and inspired many applications. However, the method solves backward stochastic differential equations (BSDEs) in a forward manner, which can not be used for optimal stopping problems that in general require running BSDE backwardly. To overcome this difficulty, a recent paper [Wang, Chen, Sudjianto, Liu and Shen, arXiv:1807.06622, 2018] proposed the backward deep BSDE method to solve the optimal stopping problem. In this paper, we provide the rigorous theory for the backward deep BSDE method. Specifically, 1. We derive the a posteriori error estimation, i.e., the error of the numerical solution can be bounded by the training loss function; and; 2. We give an upper bound of the loss function, which can be sufficiently small subject to universal approximations. We give two numerical examples, which present consistent performance with the proved theory.

翻译：最优停止问题是金融市场中的核心问题之一，广泛应用于美式和百慕大期权定价等领域。深度BSDE方法 [Han, Jentzen and E, PNAS, 115(34):8505-8510, 2018] 在求解高维正倒向随机微分方程（FBSDEs）方面展现出强大能力，并催生了众多应用。然而，该方法以正向方式求解倒向随机微分方程（BSDEs），无法直接应用于通常需要反向求解BSDE的最优停止问题。为克服这一困难，近期论文 [Wang, Chen, Sudjianto, Liu and Shen, arXiv:1807.06622, 2018] 提出了反向深度BSDE方法以解决最优停止问题。本文为该反向深度BSDE方法建立了严格的理论基础。具体而言：1. 我们推导了后验误差估计，即数值解的误差可由训练损失函数界定；2. 我们给出了损失函数的上界，该上界可通过通用逼近达到足够小。我们提供了两个数值算例，其表现与理论证明结果一致。