Numerical Approximations of Coupled Forward-Backward SPDEs

TL;DR

This paper introduces a combined finite element and machine learning approach for numerically solving complex coupled nonlinear forward-backward stochastic PDEs, extending previous methods to more general cases with proven convergence and demonstrated efficiency.

Contribution

It generalizes existing schemes to handle more complex nonlinear and nonlocal FBSPDEs with coupling, providing new proofs and numerical techniques.

Findings

Proved existence and uniqueness of solutions for the considered FBSPDEs.

Developed a finite element and deep learning-based numerical scheme.

Numerical examples show the method's efficiency in decoupled and coupled cases.

Abstract

We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. Precisely, we generalize the pioneering work of Dunst and Prohl [SIAM J. Sci. Comp., 38(2017), 2725--2755] by considering general nonlinear and nonlocal FBSPDEs with more inclusive coupling; self-contained proofs are provided and different numerical techniques for the resulting finite-dimensional equations are adopted. For such FBSPDEs, we first prove the existence and uniqueness of the strong solution as well as of the weak solution. Then the finite element method in the spatial domain leads to approximations of FBSPDEs by finite-dimensional forward-backward stochastic differential equations (FBSDEs) which are numerically computed by using some deep learning-based schemes. The convergence analysis is addressed for the spatial discretization of FBSPDEs, and the numerical examples, including both decoupled and coupled cases, indicate that our methods are quite efficient.