# Deep Learning for Reflected BSDEs: Regularization and Error Analysis

arXiv: https://arxiv.org/abs/2609.05434 · HTML: https://mlsanity.com/paper/2609.05434/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 43 references that are in the map.
- First posted on arXiv 27 Jun 2026.

## Builds on (most cited first, 19 of 19)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem](https://mlsanity.com/paper/1906.11945/) (2019)
- [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)
- [Approximation error analysis of some deep backward schemes for nonlinear PDEs](https://mlsanity.com/paper/2006.01496/) (2020)
- [Randomized Optimal Stopping Problem in Continuous time and Reinforcement Learning Algorithm](https://mlsanity.com/paper/2208.02409/) (2022)
- [Deep Neural Network Framework Based on Backward Stochastic Differential Equations for Pricing and Hedging American Options in High Dimensions](https://mlsanity.com/paper/1909.11532/) (2019)
- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems](https://mlsanity.com/paper/2210.04118/) (2022)
- [A deep solver for BSDEs with jumps](https://mlsanity.com/paper/2211.04349/) (2022)
- [Learning to Optimally Stop Diffusion Processes, with Financial Applications](https://mlsanity.com/paper/2408.09242/) (2024)
- [Deep Learning for Ranking Response Surfaces with Applications to Optimal Stopping Problems](https://mlsanity.com/paper/1901.03478/) (2019)
- [Solving stochastic optimal control problem via stochastic maximum principle with deep learning method](https://mlsanity.com/paper/2007.02227/) (2020)
- [Deep Signature Algorithm for Multi-dimensional Path-Dependent Options](https://mlsanity.com/paper/2211.11691/) (2022)
- [Convergence of a Deep BSDE solver with jumps](https://mlsanity.com/paper/2501.09727/) (2025)

## Abstract (arXiv)

Reflected backward stochastic differential equations (RBSDEs) provide a probabilistic formulation for obstacle constrained problems, but existing deep learning methods for their high dimensional solution remain limited. In this paper, we propose two deep learning schemes for RBSDEs, a deep forward scheme (DFS) and a deep backward scheme (DBS), by first reducing the reflected problem to a family of regularized BSDEs. Our main theoretical contribution concerns the DBS: we establish an explicit error bound showing that, for each fixed regularization parameter varepsilon>0, the approximation error between the DBS solution and the solution to the regularized BSDE is controlled by the associated training loss. We prove that this training loss can be controlled by the universal approximation capability of neural networks. Together, these results yield a theoretical foundation for the deep learning-based solution and complement existing analysis for forward type methods. We illustrate the framework on high dimensional American option pricing, where the reflected formulation allows us to address the continuous time exercise feature directly rather than through a Bermudan approximation. Numerical experiments demonstrate that both DFS and DBS deliver accurate solutions in high dimensions.