Picard Approximation and Martingale Integrand Estimation for Path-Dependent FBSDEs

Abstract

This paper develops a Picard iteration method for path-dependent forward-backward stochastic differential equations (FBSDEs). We construct a Picard iteration scheme, prove its convergence to the exact solution, and derive an explicit convergence rate. Although the underlying problem is conceptually straightforward, establishing rigorous convergence results in the path-dependent setting is technically demanding. Path dependence introduces a logarithmic correction: the time-discretization error bound is of order $|\pi |\log (1/|\pi |)$, compared with the order $|\pi |$ obtained in the Markovian setting, where $|\pi |$ denotes the mesh size of the time partition. We also introduce an estimator for the martingale integrand of the FBSDE that is tailored to path dependence and establish, within a Monte Carlo framework, a concentration inequality that quantifies its statistical error. The applicability of the proposed method is demonstrated in two non-Markovian derivative-pricing settings. The primary objective of this paper is to provide a theoretical convergence analysis of the Picard iteration method in the path-dependent setting and illustrate its application to option pricing, rather than to develop a faster or more computationally efficient algorithm.