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

2025-03-17

arXiv.org

AI Summary

The deep BSDE method of Han, Jentzen, and E (2018) frequently fails to converge when applied to coupled forward-backward stochastic differential equations (FBSDEs). Method: We propose a robust numerical framework comprising (1) the construction of the first multi-FBSDE equivalence family, ensuring PDE consistency of solutions, and (2) a two-stage approximation scheme integrating deep neural networks, SDE discretization, and iterative optimization of initial conditions. Contribution/Results: Our method achieves stable convergence across multiple high-dimensional, nonlinear coupled FBSDE benchmark problems—where the standard deep BSDE method diverges entirely. Empirical results demonstrate substantial improvements in convergence robustness and applicability scope, establishing a novel paradigm for efficiently solving complex coupled FBSDEs.

Technology Category

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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.

Problem

• Robust approximation of coupled FBSDEs
• Overcoming convergence issues in deep BSDE method
• Equivalent FBSDE family satisfying same PDE

Innovation

• Deep multi-FBSDE method for robust approximation
• Equivalent FBSDE family to solve convergence issues
• Two-phase algorithm for initial condition approximation