Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deep Forward-Backward Dynamic Programming Schemes for High-Dimensional Semilinear Nonlocal PDEs and FBSDE with Jumps

Wansheng Wang, Jiangtao Pan, Jie Wang, Zaijun Ye

TL;DR

This work tackles high-dimensional parabolic PIDEs and forward-backward SDEs with jumps by introducing the DFBDP algorithm, a deep learning-based dynamic programming approach that jointly approximates the solution $u(t,x)$ and the nonlocal kernel $B[u]$ with neural networks, while computing $\nabla_x u$ via numerical differentiation. The method builds on the nonlinear Feynman-Kac representation and extends the DBDP2 framework to nonlocal terms, using Euler discretization for the forward equation and policy nets $\mathcal{U}_i$, $\mathcal{G}_i$ to approximate $u$ and $U$; a rigorous convergence analysis under assumptions (H1)-(H2) yields an error bound that accounts for terminal mismatch, time discretization, and neural-network approximation errors. The theoretical results are complemented by numerical experiments in both 1D and higher dimensions across multiple Lévy measures, demonstrating accurate approximations of $u$ and its gradient with errors that remain small as dimension grows. The paper provides a practical, scalable tool for solving high-dimensional, nonlocal PDEs arising in physics and finance, with code made available for reproducibility.

Abstract

We propose a new deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and forward-backward stochastic differential equations with jumps (FBSDEJs). This novel algorithm can be viewed as an extension and generalization of the DBDP2 scheme and a dynamic programming version of the forward-backward algorithm proposed recently for high-dimensional semilinear PDEs and semilinear PIDEs, respectively. Different from the DBDP2 scheme for semilinear PDEs, our algorithm approximate simultaneously the solution and the integral kernel by deep neural networks, while the gradient of the solution is approximated by numerical differential techniques. The related error estimates for the integral kernel approximation play key roles in deriving error estimates for the novel algorithm. Numerical experiments confirm our theoretical results and verify the effectiveness of the proposed methods.

Deep Forward-Backward Dynamic Programming Schemes for High-Dimensional Semilinear Nonlocal PDEs and FBSDE with Jumps

TL;DR

This work tackles high-dimensional parabolic PIDEs and forward-backward SDEs with jumps by introducing the DFBDP algorithm, a deep learning-based dynamic programming approach that jointly approximates the solution and the nonlocal kernel with neural networks, while computing via numerical differentiation. The method builds on the nonlinear Feynman-Kac representation and extends the DBDP2 framework to nonlocal terms, using Euler discretization for the forward equation and policy nets , to approximate and ; a rigorous convergence analysis under assumptions (H1)-(H2) yields an error bound that accounts for terminal mismatch, time discretization, and neural-network approximation errors. The theoretical results are complemented by numerical experiments in both 1D and higher dimensions across multiple Lévy measures, demonstrating accurate approximations of and its gradient with errors that remain small as dimension grows. The paper provides a practical, scalable tool for solving high-dimensional, nonlocal PDEs arising in physics and finance, with code made available for reproducibility.

Abstract

We propose a new deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and forward-backward stochastic differential equations with jumps (FBSDEJs). This novel algorithm can be viewed as an extension and generalization of the DBDP2 scheme and a dynamic programming version of the forward-backward algorithm proposed recently for high-dimensional semilinear PDEs and semilinear PIDEs, respectively. Different from the DBDP2 scheme for semilinear PDEs, our algorithm approximate simultaneously the solution and the integral kernel by deep neural networks, while the gradient of the solution is approximated by numerical differential techniques. The related error estimates for the integral kernel approximation play key roles in deriving error estimates for the novel algorithm. Numerical experiments confirm our theoretical results and verify the effectiveness of the proposed methods.
Paper Structure (12 sections, 3 theorems, 103 equations, 36 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 3 theorems, 103 equations, 36 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Deep learning-based schemes for semilinear PIDEs and decoupled FBSDEJs
  3. Semilinear PIDEs and FBSDEJs
  4. Deep learning (DL) and function approximation
  5. The DFBDP scheme
  6. Convergence analysis for DFBDP scheme
  7. Assumptions
  8. Main convergence results
  9. Numerical examples
  10. One-dimensional problem
  11. High-dimensional problems
  12. Conclusions

Key Result

Lemma 3.1

\newlabelle2.1 Let $Z^1, Z^2\in L^{2}_{W}(\mathbb{R}^{d})$ and $U^1, U^2\in L^{2}_{\mu}(\mathbb{R})$, we have and

Figures (36)

  • Figure 4.1: Uniform distribution
  • Figure 4.2: Normal distribution
  • Figure 4.3: Exponential distribution
  • Figure 4.4: Bernoulli distribution
  • Figure 4.6: $u(0,x)$
  • ...and 31 more figures

Theorems & Definitions (4)

  • Lemma 3.1: R2005
  • Lemma 3.2: Martingale representation theorem R2005
  • Theorem 3.3: Convergence of DFBDP scheme
  • proof