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A discretization scheme for path-dependent FBSDEs and PDEs

Jiuk Jang, Hyungbin Park

TL;DR

This work develops a numerical framework for path-dependent FBSDEs and PPDEs by introducing a Picard-type iteration that accommodates full path dependence. A novel, robust estimator for the martingale integrand is paired with a concentration inequality to quantify Monte Carlo errors, enabling reliable, basis-free estimation in non-Markovian settings. The authors also formulate a supervised learning approach using neural networks to solve PPDEs, backed by a universal-approximation result in the path-dependent context. Together, these components produce a fully implementable scheme applicable to a broad class of path-dependent problems with provable convergence and error bounds, and they demonstrate practical performance on PPDEs and lookback options.

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.

A discretization scheme for path-dependent FBSDEs and PDEs

TL;DR

This work develops a numerical framework for path-dependent FBSDEs and PPDEs by introducing a Picard-type iteration that accommodates full path dependence. A novel, robust estimator for the martingale integrand is paired with a concentration inequality to quantify Monte Carlo errors, enabling reliable, basis-free estimation in non-Markovian settings. The authors also formulate a supervised learning approach using neural networks to solve PPDEs, backed by a universal-approximation result in the path-dependent context. Together, these components produce a fully implementable scheme applicable to a broad class of path-dependent problems with provable convergence and error bounds, and they demonstrate practical performance on PPDEs and lookback options.

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.
Paper Structure (15 sections, 14 theorems, 115 equations, 3 tables)

This paper contains 15 sections, 14 theorems, 115 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Preliminaries
  4. Functional Itô calculus
  5. Path-dependent FBSDEs
  6. Euler scheme for path-dependent SDEs
  7. Picard iteration method
  8. Smooth approximation
  9. Main results
  10. Second-order parabolic path-dependent PDEs
  11. Error analysis for the martingale integrand
  12. Numerical simulations
  13. Path-dependent PDEs
  14. Lookback options
  15. Conclusion

Key Result

Proposition 2.1

Let Assumption asmsde hold. Then for any $t \in [0, T]$ and $\gamma \in D([0,T],\mathbb{R}^{d})$, there exists a unique solution $X^{\gamma_{t}}$ to eqn:psde in $\mathbb{S}^2(0,T;\mathbb{R}^d)$. Moreover, there exists a constant $C>0$ such that for all $t,t' \in [0, T]$, $\gamma,\gamma' \in D([0,T],\mathbb{R}^{d})$, and $s,s'\in [t,T],$

Theorems & Definitions (23)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 13 more