The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEs

ArXiv ID: 2503.13193 “View on arXiv”

Authors: Unknown

Abstract

We introduce the deep multi-FBSDE method for robust approximation of coupled forward-backward stochastic differential equations (FBSDEs), focusing on cases where the deep BSDE method of Han, Jentzen, and E (2018) fails to converge. To overcome the convergence issues, we consider a family of FBSDEs that are equivalent to the original problem in the sense that they satisfy the same associated partial differential equation (PDE). Our algorithm proceeds in two phases: first, we approximate the initial condition for the FBSDE family, and second, we approximate the original FBSDE using the initial condition approximated in the first phase. Numerical experiments show that our method converges even when the standard deep BSDE method does not.

Keywords: Forward-Backward Stochastic Differential Equations (FBSDEs), Deep Learning, Partial Differential Equations (PDEs), Monte Carlo Methods, High-Dimensional Approximation, Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, employing advanced stochastic calculus, FBSDE theory, and rigorous numerical analysis, but lacks backtest-ready code or heavy implementation focus, showing only controlled numerical experiments.

  flowchart TD
    A["Research Goal<br>Robust Deep Learning for Coupled FBSDEs"] --> B["Data/Input<br>FBSDE-PDE System Parameters"]
    B --> C["Methodology: Two-Phase Training"]
    C --> D["Phase 1: Approximate<br>Initial Condition via Equivalent FBSDE Family"]
    D --> E["Phase 2: Approximate<br>Original FBSDE using Initial Condition"]
    E --> F["Computational Process<br>Monte Carlo Simulation & Neural Network Optimization"]
    F --> G["Key Finding<br>Convergence Achieved where Standard Deep BSDE Method Fails"]