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

TL;DR

This paper introduces a deep learning algorithm for high-dimensional semilinear PDEs and FBSDEs with jumps, extending existing schemes by approximating solutions and integral kernels with neural networks.

Contribution

The proposed algorithm generalizes previous methods by simultaneously approximating solutions and integral kernels using deep neural networks, with theoretical error estimates.

Findings

Numerical experiments confirm the effectiveness of the algorithm.

The method achieves accurate approximations for high-dimensional problems.

Error estimates support the theoretical convergence of the approach.

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.