# A Posteriori Error Analysis for Decoupled Neural Approximations of Fully Coupled FBSDEs with Control Mismatch

arXiv: https://arxiv.org/abs/2606.29474 · HTML: https://mlsanity.com/paper/2606.29474/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 31 references that are in the map.
- First posted on arXiv 28 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [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)
- [Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis](https://mlsanity.com/paper/1812.04300/) (2018)
- [Numerical resolution of McKean-Vlasov FBSDEs using neural networks](https://mlsanity.com/paper/1909.12678/) (2019)
- [Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications](https://mlsanity.com/paper/1812.05916/) (2018)
- [Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learning](https://mlsanity.com/paper/1907.05327/) (2019)
- [Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems](https://mlsanity.com/paper/2210.04118/) (2022)
- [A posteriori error estimates for fully coupled McKean-Vlasov forward-backward SDEs](https://mlsanity.com/paper/2007.07731/) (2020)
- [Convergence of a Deep BSDE solver with jumps](https://mlsanity.com/paper/2501.09727/) (2025)
- [The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEs](https://mlsanity.com/paper/2503.13193/) (2025)

## Abstract (arXiv)

This paper develops an a posteriori error analysis framework for decoupled neural approximations of fully coupled forward--backward stochastic differential equations (FBSDEs). It provides an a posteriori error-analysis for the idealized discrete adapted trajectory. The main feature of the proposed formulation is the use of an auxiliary control process in the forward coefficients, which may differ from the backward component approximated by the neural network. This decoupling is useful in practical deep learning implementations, but it creates a control mismatch that must be included in the error analysis. We first establish a continuous-time stability estimate for fully coupled FBSDEs under perturbations of the drift, diffusion, generator, terminal condition, and auxiliary control input. We then transfer this estimate to the discrete-time setting and derive computable a posteriori error bounds depending only on the terminal defect, the pathwise residual, and the control mismatch. When the auxiliary control is identified with the backward approximation, the mismatch term vanishes and the bound reduces to the standard two-term form. Numerical experiments on a linear--quadratic FBSDE with an explicit reference solution and a multidimensional Burgers-type FBSDE without a reference solution illustrate the diagnostic role of the proposed indicators and the contribution of the mismatch penalty to the consistency and reproducibility of the numerical approximations.