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FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equations

Zaijun Ye, Wansheng Wang

TL;DR

This work develops Forward-Backward Stochastic Jump Neural Networks (FBSJNN) to solve partial integro-differential equations by transforming PIDEs into FBSDEJs via a nonlinear Feynman-Kac representation and discretizing with Euler-Maruyama. A single neural network approximates the backward component and, through Taylor expansion, the non-local jump integral, enabling efficient training with automatic differentiation. The authors prove consistency: the discretization error scales with $\Delta t$ and the neural network approximation error $\epsilon^{\mathcal{Y}}$, and show that as the network width grows (universal approximation), the method converges to the true FBSDEJ solution. Numerical experiments across 1D and high-dimensional PIDEs and jump-dominated models demonstrate rapid convergence and sub-1e-2 level relative errors, including a high-dimensional test with $d=100$ and a multi-layer financial model. Overall, FBSJNN offers a theoretically grounded, parameter-efficient, scalable framework for high-dimensional stochastic PIDEs with jumps, supported by rigorous consistency results and strong empirical performance.

Abstract

We propose a novel framework for solving a class of Partial Integro-Differential Equations (PIDEs) and Forward-Backward Stochastic Differential Equations with Jumps (FBSDEJs) through a deep learning-based approach. This method, termed the Forward-Backward Stochastic Jump Neural Network (FBSJNN), is both theoretically interpretable and numerically effective. Theoretical analysis establishes the convergence of the numerical scheme and provides error estimates grounded in the universal approximation properties of neural networks. In comparison to existing methods, the key innovation of the FBSJNN framework is that it uses a single neural network to approximate both the solution of the PIDEs and the non-local integral, leveraging Taylor expansion for the latter. This enables the method to reduce the total number of parameters in FBSJNN, which enhances optimization efficiency. Numerical experiments indicate that the FBSJNN scheme can obtain numerical solutions with a relative error on the scale of $10^{-3}$.

FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equations

TL;DR

This work develops Forward-Backward Stochastic Jump Neural Networks (FBSJNN) to solve partial integro-differential equations by transforming PIDEs into FBSDEJs via a nonlinear Feynman-Kac representation and discretizing with Euler-Maruyama. A single neural network approximates the backward component and, through Taylor expansion, the non-local jump integral, enabling efficient training with automatic differentiation. The authors prove consistency: the discretization error scales with and the neural network approximation error , and show that as the network width grows (universal approximation), the method converges to the true FBSDEJ solution. Numerical experiments across 1D and high-dimensional PIDEs and jump-dominated models demonstrate rapid convergence and sub-1e-2 level relative errors, including a high-dimensional test with and a multi-layer financial model. Overall, FBSJNN offers a theoretically grounded, parameter-efficient, scalable framework for high-dimensional stochastic PIDEs with jumps, supported by rigorous consistency results and strong empirical performance.

Abstract

We propose a novel framework for solving a class of Partial Integro-Differential Equations (PIDEs) and Forward-Backward Stochastic Differential Equations with Jumps (FBSDEJs) through a deep learning-based approach. This method, termed the Forward-Backward Stochastic Jump Neural Network (FBSJNN), is both theoretically interpretable and numerically effective. Theoretical analysis establishes the convergence of the numerical scheme and provides error estimates grounded in the universal approximation properties of neural networks. In comparison to existing methods, the key innovation of the FBSJNN framework is that it uses a single neural network to approximate both the solution of the PIDEs and the non-local integral, leveraging Taylor expansion for the latter. This enables the method to reduce the total number of parameters in FBSJNN, which enhances optimization efficiency. Numerical experiments indicate that the FBSJNN scheme can obtain numerical solutions with a relative error on the scale of .

Paper Structure

This paper contains 16 sections, 7 theorems, 81 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Forward-backward stochastic jump neural networks
  3. PIDEs
  4. FBSDEJs
  5. Time dicretization of FBSDEJs
  6. Forward-backward stochastic jump neural network
  7. Consistency Analysis
  8. Preliminaries
  9. Consistency of the FBSJNN
  10. Numerical Examples
  11. One-dimensional pure jump equation
  12. One-dimensional partial integro-differential equation
  13. High-dimensional partial integro-differential equation
  14. High-dimensional Black-Scholes-Barenblatt equations
  15. Time discretization error test
  16. ...and 1 more sections

Key Result

Lemma 2.1

There exists a unique solution $(X^x_t,Y^x_t,Z^x_t,I^x_t)$ to the FBSDEJ eqn:FBSDEJ under Assumptions assumption:A1 - assumption:A5.

Figures (6)

  • Figure 1: Forward-Backward Stochastic Jump Neural Network.
  • Figure 2: Solution of the one-dimensional pure jump equation.
  • Figure 3: Solution of the one-dimensional partial integro-differential equation.
  • Figure 4: Heat map of absolute errors in spatio-temporal region.
  • Figure 5: Solution of the high-dimensional partial integro-differential equations.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Lemma 2.1: Theorem 4.1.3 and Proposition 4.1.1 in delongBackwardStochasticDifferential2013
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1: Martingale Representation Theorem, situTheoryStochasticDifferential2005
  • Lemma 3.2: Conditional Itô isometry, situTheoryStochasticDifferential2005
  • Lemma 3.3: $L^2$ regularity of $Y$, $Z$ and $I$, bouchardDiscretetimeApproximationDecoupled2008castroDeepLearningSchemes2022
  • Lemma 3.4: Square-integral property of $f$, castroDeepLearningSchemes2022
  • Lemma 3.5: Universal Approximation Theorem, hornikMultilayerFeedforwardNetworks1989stinchcombeUniversalApproximationUsing1989
  • ...and 2 more