Deep learning schemes for parabolic nonlocal integro-differential equations

Abstract

AbstractIn this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre by Lius and Lius (Comm PDE 32(8):1245–1260, 2007). Based on the work by Hure, Pham and Warin by Hure et al. (Math Comp 89:1547–1579, 2020), we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on Lévy processes and a new neural network approximation of the nonlocal part to overcome the lack of a suitable good approximation of the nonlocal part of the solution.