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

arXiv: https://arxiv.org/abs/2210.04118 · HTML: https://mlsanity.com/paper/2210.04118/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 8 Oct 2022.

## Builds on (most cited first, 8 of 8)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (2018)
- [Optimal Stopping via Randomized Neural Networks](https://mlsanity.com/paper/2104.13669/) (2021)
- [Deep Fictitious Play for Stochastic Differential Games](https://mlsanity.com/paper/1903.09376/) (2019)
- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [Convergence of Deep Fictitious Play for Stochastic Differential Games](https://mlsanity.com/paper/2008.05519/) (2020)
- [Deep Learning for Ranking Response Surfaces with Applications to Optimal Stopping Problems](https://mlsanity.com/paper/1901.03478/) (2019)

## Built on by (most cited first, 9 of 9)

- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Deep Signature Algorithm for Multi-dimensional Path-Dependent Options](https://mlsanity.com/paper/2211.11691/) (2022)
- [Simultaneous upper and lower bounds of American-style option prices with hedging via neural networks](https://mlsanity.com/paper/2302.12439/) (2023)
- [Deep Neural Operator Learning for Probabilistic Models](https://mlsanity.com/paper/2511.07235/) (2025)
- [Deep Learning for the Multiple Optimal Stopping Problem](https://mlsanity.com/paper/2512.22961/) (2025)
- [Deep Generative Modeling with Backward Stochastic Differential Equations](https://mlsanity.com/paper/2304.04049/) (2023)
- [Continuous-time Optimal Stopping through Deep Reinforcement Learning](https://mlsanity.com/paper/2606.17545/) (2026)
- [A Posteriori Error Analysis for Decoupled Neural Approximations of Fully Coupled FBSDEs with Control Mismatch](https://mlsanity.com/paper/2606.29474/) (2026)
- [Deep Learning for Reflected BSDEs: Regularization and Error Analysis](https://mlsanity.com/paper/2609.05434/) (2026)

## Abstract (arXiv)

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.