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Generalized convergence of the deep BSDE method: a step towards fully-coupled FBSDEs and applications in stochastic control

Balint Negyesi, Zhipeng Huang, Cornelis W. Oosterlee

TL;DR

This work generalizes the convergence analysis of the deep BSDE method to fully coupled FBSDEs where the forward drift can depend on $Z$, extending beyond prior results that only allowed $Y$-coupling. It derives a posteriori error bounds showing the discretization error plus the terminal loss controls the overall approximation error, and introduces new lemmas to handle $Z$-coupling in the forward equation. The main result identifies contraction conditions, via limits $ar{B}$ and $ar{A}$, under which convergence is guaranteed when $ar{B}$ and/or $ar{A}$ are strictly less than 1; these conditions are verifiable for any given equation and interpreted in regimes like small time horizon or weak coupling. Numerical experiments in high dimensions demonstrate the theory: fully coupled examples converge only when the contraction conditions hold, while DP formulations with strong $Z$-coupling may diverge unless the coupling strength is reduced or SMP-based formulations are used. Overall, the paper provides practical guidelines for when the deep BSDE method converges in stochastic control problems and explains observed non-convergence phenomena with a rigorous contraction-analysis framework.

Abstract

We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to drift coefficients that may also depend on Z, and give sufficient conditions for convergence under standard assumptions. The resulting conditions are directly verifiable for any equation. Consequently, unlike in earlier theory, our convergence analysis enables the treatment of FBSDEs stemming from stochastic optimal control problems. In particular, we provide a theoretical justification for the non-convergence of the deep BSDE method observed in recent literature, and present direct guidelines for when convergence can be guaranteed in practice. Our theoretical findings are supported by several numerical experiments in high-dimensional settings.

Generalized convergence of the deep BSDE method: a step towards fully-coupled FBSDEs and applications in stochastic control

TL;DR

This work generalizes the convergence analysis of the deep BSDE method to fully coupled FBSDEs where the forward drift can depend on , extending beyond prior results that only allowed -coupling. It derives a posteriori error bounds showing the discretization error plus the terminal loss controls the overall approximation error, and introduces new lemmas to handle -coupling in the forward equation. The main result identifies contraction conditions, via limits and , under which convergence is guaranteed when and/or are strictly less than 1; these conditions are verifiable for any given equation and interpreted in regimes like small time horizon or weak coupling. Numerical experiments in high dimensions demonstrate the theory: fully coupled examples converge only when the contraction conditions hold, while DP formulations with strong -coupling may diverge unless the coupling strength is reduced or SMP-based formulations are used. Overall, the paper provides practical guidelines for when the deep BSDE method converges in stochastic control problems and explains observed non-convergence phenomena with a rigorous contraction-analysis framework.

Abstract

We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to drift coefficients that may also depend on Z, and give sufficient conditions for convergence under standard assumptions. The resulting conditions are directly verifiable for any equation. Consequently, unlike in earlier theory, our convergence analysis enables the treatment of FBSDEs stemming from stochastic optimal control problems. In particular, we provide a theoretical justification for the non-convergence of the deep BSDE method observed in recent literature, and present direct guidelines for when convergence can be guaranteed in practice. Our theoretical findings are supported by several numerical experiments in high-dimensional settings.
Paper Structure (11 sections, 5 theorems, 58 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 5 theorems, 58 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The deep BSDE algorithm
  3. Convergence analysis
  4. Interpretation of the conditions in theorem \ref{['thm:estimate2']}
  5. Numerical experiments
  6. Example 1
  7. Example 2
  8. Non-convergence of the deep BSDE method
  9. Stochastic control via DP or SMP
  10. Conclusion
  11. Acknowledgement

Key Result

Theorem 1

Under assumptions assumption:lipschitz, assumption:bounded and assumption:pde, the FBSDE eq:FBSDE has a unique solution $(X, Y, Z)$, and it holds that for $t\in [0, T]$,

Figures (3)

  • Figure 1: Example \ref{['sec:ex1']}. $T=0.25, X_0=(\pi/4,\dots,\pi/4)$.
  • Figure 2: Example \ref{['sec:ex2:1']}. Comparison of the Deep BSDE method on \ref{['eq:ex2:dp']} between coefficients as in \ref{['eq:ex2:lq_coefficients']} (original) and $M_u$ replaced by $M_u/150$ (rescaled). $T=1/2, X_0=(0.1, \dots, 0.1)$.
  • Figure 3: Example \ref{['sec:ex2:2']}. Comparison between the FBSDEs derived via dynamic programming \ref{['eq:ex2:dp']} and the stochastic maximum principle \ref{['eq:ex2:smp']} approximated by the deep BSDE method. Coefficients as in \ref{['eq:ex2:lq_coefficients']}, $T=10^{-3}, X_0=(0.1, \dots, 0.1)$.

Theorems & Definitions (14)

  • Theorem 1: Feynman-Kac
  • Remark 1
  • Theorem 2: Convergence of the implicit scheme
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 3
  • Theorem 3: Convergence of the deep BSDE method
  • ...and 4 more