Temporal Difference Learning for High-Dimensional PIDEs with Jumps

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C349-C368, August 2024.
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 Lévy processes and construct a corresponding reinforcement learning model. To simulate the entire process, we use deep neural networks to represent the solutions and nonlocal terms of the equations. Subsequently, we train the networks using the temporal difference error, the termination condition, and properties of the nonlocal terms as the loss function. The relative error of the method reaches [math] in 100-dimensional experiments and [math] 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.