  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.
