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 Beck, Becker, Cheridito, Jentzen, Neufeld (2021) and Germain, Pham, Warin (2022) to PIDEs. Our main contribution is a convergence analysis of the scheme. Moreover we discuss several test case studies to show the viability of our approach.