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A Deep BSDE approximation of nonlinear integro-PDEs with unbounded nonlocal operators

Espen Robstad Jakobsen, Sehail Mazid

TL;DR

The paper develops a Deep BSDE method to approximate solutions of nonlinear integro-PDEs with unbounded nonlocal operators, leveraging a forward–backward SDE representation driven by Brownian motion and a jump measure. It introduces a jump-diffusion approximation for small jumps, a jump-corrected Euler discretization, and neural network regression to learn the backward components, including the $U$-term associated with jumps. A detailed convergence analysis is provided, showing that the DeepBSDE scheme converges to the (nonlocal) FBSDE solution with explicit error terms arising from time discretization, jump-space quadrature, projection of conditional expectations, and neural-network approximation. The framework addresses the curse of dimensionality and is designed to handle infinite activity in the Lévy measure, providing sharp error bounds and conditions under which convergence holds. Overall, the work extends deep BSDE methods to high-dimensional PIDEs with unbounded nonlocal operators and offers a rigorous route to reliable numerical approximation in complex stochastic control and mean-field game contexts.

Abstract

Machine learning for partial differential equations (PDEs) is a hot topic. In this paper we introduce and analyse a Deep BSDE scheme for nonlinear integro-PDEs with unbounded nonlocal operators -problems arising in e.g. stochastic control and games involving infinite activity jump-processes. The scheme is based on a stochastic forward-backward SDE representation of the solution of the PDE and (i) approximation of small jumps by a Gaussian process, (ii) simulation of the forward part, and (iii) a neural net regression for the backward part. Unlike grid-based schemes, it does not suffer from the curse of dimensionality and is therefore suitable for high dimensional problems. The scheme is designed to be convergent even in the infinite activity/unbounded nonlocal operator case. A full convergence analysis is given and constitutes the main part of the paper.

A Deep BSDE approximation of nonlinear integro-PDEs with unbounded nonlocal operators

TL;DR

The paper develops a Deep BSDE method to approximate solutions of nonlinear integro-PDEs with unbounded nonlocal operators, leveraging a forward–backward SDE representation driven by Brownian motion and a jump measure. It introduces a jump-diffusion approximation for small jumps, a jump-corrected Euler discretization, and neural network regression to learn the backward components, including the -term associated with jumps. A detailed convergence analysis is provided, showing that the DeepBSDE scheme converges to the (nonlocal) FBSDE solution with explicit error terms arising from time discretization, jump-space quadrature, projection of conditional expectations, and neural-network approximation. The framework addresses the curse of dimensionality and is designed to handle infinite activity in the Lévy measure, providing sharp error bounds and conditions under which convergence holds. Overall, the work extends deep BSDE methods to high-dimensional PIDEs with unbounded nonlocal operators and offers a rigorous route to reliable numerical approximation in complex stochastic control and mean-field game contexts.

Abstract

Machine learning for partial differential equations (PDEs) is a hot topic. In this paper we introduce and analyse a Deep BSDE scheme for nonlinear integro-PDEs with unbounded nonlocal operators -problems arising in e.g. stochastic control and games involving infinite activity jump-processes. The scheme is based on a stochastic forward-backward SDE representation of the solution of the PDE and (i) approximation of small jumps by a Gaussian process, (ii) simulation of the forward part, and (iii) a neural net regression for the backward part. Unlike grid-based schemes, it does not suffer from the curse of dimensionality and is therefore suitable for high dimensional problems. The scheme is designed to be convergent even in the infinite activity/unbounded nonlocal operator case. A full convergence analysis is given and constitutes the main part of the paper.
Paper Structure (14 sections, 15 theorems, 109 equations, 1 algorithm)

This paper contains 14 sections, 15 theorems, 109 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nonlocal nonlinear partial differential equations.
  3. Connection to forward-backward stochastic differential equations with jumps
  4. Notation and Preliminaries.
  5. Forward-backward stochastic differential equation with jumps.
  6. Deep FBSDE approximation of PIDE (\ref{['PIDE2']}).
  7. Discretization of the FBSDE
  8. Neural network architecture
  9. Deep learning approximations.
  10. Intermediate approximation and errors.
  11. Convergence of the DeepBSDE scheme
  12. Proof of convergence, Theorem \ref{['theo-convergence1']}.
  13. Proof of Theorem \ref{['strong convergence-app']}.
  14. Proof of (\ref{['strong error-U']})

Key Result

Theorem 2.2

Assume (H1) and (H2). Then there exists a unique viscosity solution $u$ of (PIDE). Moreover, there exists $C> 0$ such that for $x,x'\in\mathbb{R}^q$ and $t\in[0,T)$,

Theorems & Definitions (29)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 3.1
  • Definition 3.2
  • Corollary 3.3
  • Remark 3.4
  • Theorem 3.5
  • ...and 19 more