Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems

TL;DR

This paper provides a rigorous theoretical foundation for the backward deep BSDE method, used to solve optimal stopping problems like American options, by establishing error bounds and demonstrating its effectiveness through numerical examples.

Contribution

The paper develops a posteriori error estimates and bounds for the backward deep BSDE method, enhancing its theoretical understanding and practical reliability.

Findings

Error of the numerical solution is bounded by the training loss.

An upper bound of the loss function can be made small with universal approximation.

Numerical examples confirm the theoretical results.

Abstract

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.