A discretization scheme for path-dependent FBSDEs and PDEs

TL;DR

This paper introduces a new numerical scheme for solving path-dependent FBSDEs and PDEs, featuring a Picard iteration, a novel martingale estimator, and a neural network-based approach, with proven convergence and error bounds.

Contribution

It presents a novel estimator for the martingale integrand in path-dependent FBSDEs and a fully implementable neural network algorithm for related PDEs, with convergence guarantees.

Findings

Convergence of the Picard iteration for path-dependent FBSDEs.

A concentration inequality for the martingale estimator.

An effective neural network-based solver for path-dependent PDEs.

Abstract

This study develops a numerical scheme for path-dependent FBSDEs and PDEs. We introduce a Picard iteration method for solving path-dependent FBSDEs, prove its convergence to the true solution, and establish its rate of convergence. A key contribution of our approach is a novel estimator for the martingale integrand in the FBSDE, specifically designed to handle path-dependence more reliably than existing methods. We derive a concentration inequality that quantifies the statistical error of this estimator in a Monte Carlo framework. Based on these results, we investigate a supervised learning method with neural networks for solving path-dependent PDEs. The proposed algorithm is fully implementable and adaptable to a broad class of path-dependent problems.