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

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.