🤖 AI Summary
This work addresses the control mismatch arising in decoupled neural approximations of fully coupled forward–backward stochastic differential equations (FBSDEs), where inconsistency between the auxiliary control and the backward component compromises solution accuracy, alongside the lack of reliable a posteriori error estimates. For the first time, the control mismatch is explicitly incorporated into the a posteriori error analysis framework for neural approximations. By introducing an auxiliary control process to satisfy the decoupling requirement inherent in deep learning implementations, and leveraging tools from stochastic analysis and numerical FBSDE theory, the authors establish stability estimates in both continuous and discrete time, yielding computable error bounds that depend solely on terminal defects, pathwise residuals, and the mismatch term. Numerical experiments on a linear-quadratic FBSDE with an explicit solution and a high-dimensional Burgers-type FBSDE without a reference solution demonstrate the effectiveness of the proposed error indicators, showing that penalizing control mismatch significantly enhances the consistency and reproducibility of numerical solutions.
📝 Abstract
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.