Temporal Difference Learning for High-Dimensional PIDEs with Jumps

TL;DR

This paper introduces a deep learning approach using temporal difference learning to efficiently solve high-dimensional PIDEs with jumps, achieving high accuracy and robustness in complex stochastic models.

Contribution

The paper presents a novel deep reinforcement learning framework for high-dimensional PIDEs with jumps, incorporating Levy processes and neural networks for solution approximation.

Findings

Achieves relative error of O(10^{-3}) in 100-dimensional problems

Attains relative error of O(10^{-4}) in 1D pure jump problems

Demonstrates low computational cost and robustness across different jump scenarios

Abstract

In this paper, we propose a deep learning framework for solving high-dimensional partial integro-differential equations (PIDEs) based on the temporal difference learning. We introduce a set of Levy processes and construct a corresponding reinforcement learning model. To simulate the entire process, we use deep neural networks to represent the solutions and non-local terms of the equations. Subsequently, we train the networks using the temporal difference error, termination condition, and properties of the non-local terms as the loss function. The relative error of the method reaches O(10^{-3}) in 100-dimensional experiments and O(10^{-4}) in one-dimensional pure jump problems. Additionally, our method demonstrates the advantages of low computational cost and robustness, making it well-suited for addressing problems with different forms and intensities of jumps.