Deep learning numerical methods for high-dimensional fully nonlinear PIDEs and coupled FBSDEs with jumps

TL;DR

This paper introduces a deep learning algorithm that effectively solves high-dimensional PIDEs and FBSDEJs with jumps, leveraging neural networks to approximate gradients and integral kernels, with proven convergence and demonstrated efficiency.

Contribution

The paper presents a novel deep learning framework for high-dimensional PIDEs and FBSDEJs with jumps, including error analysis and numerical validation.

Findings

The algorithm accurately solves high-dimensional PIDEs and FBSDEJs with jumps.

Error estimates and convergence proofs support the method's reliability.

Numerical examples demonstrate the method's efficiency and effectiveness.

Abstract

We propose a deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and high-dimensional forward-backward stochastic differential equations with jumps (FBSDEJs), where the jump-diffusion process are derived by a Brownian motion and an independent compensated Poisson random measure. In this novel algorithm, a pair of deep neural networks for the approximations of the gradient and the integral kernel is introduced in a crucial way based on deep FBSDE method. To derive the error estimates for this deep learning algorithm, the convergence of Markovian iteration, the error bound of Euler time discretization, and the simulation error of deep learning algorithm are investigated. Two numerical examples are provided to show the efficiency of this proposed algorithm.