A Parallel Four Step Domain Decomposition Scheme for Coupled Forward Backward Stochastic Differential Equations

TL;DR

This paper introduces a parallel domain decomposition scheme for solving coupled forward-backward stochastic differential equations, enabling efficient parallel computation with proven convergence.

Contribution

It reconstructs the Four Step Scheme with new conditions and integrates domain decomposition methods, providing a novel parallel approach for FBSDEs with convergence guarantees.

Findings

Scheme enables parallel computation of FBSDEs.

Convergence of the scheme is rigorously proven.

Applicable to systems of quasilinear parabolic equations.

Abstract

Motivated by the idea of imposing paralleling computing on solving stochastic differential equations (SDEs), we introduce a new Domain Decomposition Scheme to solve forward-backward stochastic differential equations (FBSDEs) parallely. We reconstruct the Four Step Scheme in {MaProtterYong:1994:SFB} with some different conditions and then associate it with the idea of Domain Decomposition Methods. We also introduce a new technique to prove the convergence of Domain Decomposition Methods for systems of quasilinear parabolic equations and use it to prove the convergence of our scheme for the FBSDEs.