# An Optimization-Based Framework for Solving Forward-Backward Stochastic Differential Equations: Convergence Analysis and Error Bounds

arXiv: https://arxiv.org/abs/2507.15234 · HTML: https://mlsanity.com/paper/2507.15234/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 12 of its 31 references that are in the map.
- First posted on arXiv 21 Jul 2025.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Recent Advances in Reinforcement Learning in Finance](https://mlsanity.com/paper/2112.04553/) (2021)
- [Policy Gradient and Actor-Critic Learning in Continuous Time and Space: Theory and Algorithms](https://mlsanity.com/paper/2111.11232/) (2021)
- [Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space: A Martingale Approach](https://mlsanity.com/paper/2108.06655/) (2021)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (2018)
- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [Deep BSDE-ML Learning and Its Application to Model-Free Optimal Control](https://mlsanity.com/paper/2201.01318/) (2022)
- [Probabilistic Framework of Howard's Policy Iteration: BML Evaluation and Robust Convergence Analysis](https://mlsanity.com/paper/2210.07473/) (2022)

## Abstract (arXiv)

Forward-backward stochastic differential equations have recently become a key focus in the computational field, and their role in continuous-time stochastic optimal control and reinforcement learning has grown increasingly prominent. In this paper, we develop an optimization-based framework for solving coupled forward-backward stochastic differential equations, which naturally arise in stochastic optimal control through the stochastic maximum principle and related Hamiltonian systems. We introduce an integral-form objective function and prove its equivalence to the error between consecutive Picard iterates. Our convergence analysis establishes that minimizing this objective generates sequences that converge to the true solution. We provide explicit upper and lower bounds that relate the objective value to the error between trial and exact solutions. We validate the proposed objective and its theoretical interpretation using two analytical test cases, and further illustrate its numerical applicability on a nonlinear stochastic optimal control problem with up to 1000 dimensions.