Convergence Analysis of the Deep Splitting Scheme: the Case of Partial Integro-Differential Equations and the associated FBSDEs with Jumps

High-dimensional parabolic partial integro-differential equations (PIDEs) appear in many applications in insurance and finance. Existing numerical methods suffer from the curse of dimensionality or provide solutions only for a given space-time point. This gave rise to a growing literature on deep learning based methods for solving partial differential equations; results for integro-differential equations on the other hand are scarce. In this paper we consider an extension of the deep splitting scheme due to arXiv:1907.03452 and arXiv:2006.01496v3 to PIDEs. Our main contribution is an analysis of the approximation error which yields convergence rates in terms of the number of neurons for shallow neural networks. Moreover we discuss several test case studies to show the viability of our approach.