Convergence of the Deep BSDE Method for Coupled FBSDEs
This provides a theoretical guarantee for a numerical method used in high-dimensional financial modeling and PDE solving, but it is incremental as it extends prior work to coupled cases.
The paper tackles the theoretical foundation of the deep BSDE method for solving high-dimensional coupled forward-backward stochastic differential equations (FBSDEs), proving that the error converges to zero with neural network approximations and demonstrating accuracy through numerical results.
The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs.