Paper: arXiv 2308.07029

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents highly advanced mathematical concepts such as functional Itô calculus, path-dependent FBSDEs, and Picard iteration convergence proofs, reflecting high mathematical density. While the method is theoretically backed with a neural network approach and a Monte Carlo estimator, the excerpt emphasizes theoretical proofs and algorithmic descriptions over concrete backtesting, datasets, or empirical performance metrics.

Research Flowchart

  flowchart TD
  A["Research Goal: Develop Numerical Scheme<br>for Path-dependent FBSDEs/PDEs"] --> B{"Methodology"}
  
  B --> C["Picard Iteration<br>for Path-dependent FBSDEs"]
  B --> D["Novel Martingale Integrand<br>Estimator"]
  B --> E["Monte Carlo Framework"]
  
  C --> F["Proof of Convergence<br>and Rate"]
  D --> G["Concentration Inequality<br>for Statistical Error"]
  E --> H["Supervised Learning<br>with Neural Networks"]
  
  F & G & H --> I["Key Outcomes:<br>1. Convergent Numerical Scheme<br>2. Error Quantification<br>3. Fully Implementable Algorithm"]