An Optimization-Based Framework for Solving Forward-Backward Stochastic Differential Equations: Convergence Analysis and Error Bounds
On the map: Deep reinforcement learning methods → RL theory and sample complexity
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.
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.
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References · 12
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- 2026The Bias of Nonlinear Two-Time-scale Stochastic Approximation under Constant Step-Sizes
- 2026Discretization error from regularized Reinforcement Learning to continuous-time stochastic control
- 2026Lyapunov Stability of Stochastic Vector Optimization: Theory and Numerical Implementation ↗
- 2024Backward Stochastic Control System with Entropy Regularization ↗
- 2026Learning-Based Stochastic Optimal Control with Infinite-Horizon Probabilistic Constraints