Multilevel Picard approximation algorithm for semilinear partial integro-differential equations and its complexity analysis

Abstract

In this paper we introduce a multilevel Picard approximation algorithm for semilinear parabolic partial integro-differential equations (PIDEs). We prove that the numerical approximation scheme converges to the unique viscosity solution of the PIDE under consideration. To that end, we derive a Feynman-Kac representation for the unique viscosity solution of the semilinear PIDE, extending the classical Feynman-Kac representation for linear PIDEs. Furthermore, we show that the algorithm does not suffer from the curse of dimensionality, i.e. the computational complexity of the algorithm is bounded polynomially in the dimension d and the prescribed reciprocal of the accuracy ε.Key words and phrases. Multilevel Picard approximation, Nonlinear PIDE, overcoming the curse of dimensionality, Monte Carlo methods, Feynman-Kac representation.Financial support by the Nanyang Assistant Professorship Grant (NAP Grant) Machine Learning based Algorithms in Finance and Insurance is gratefully acknowledged.

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

“…However, they do not study the convergence properties of their scheme, and they assume that the driver f is independent of D x u, which excludes many relevant control problems. Finally, Neufeld and Wu [21] consider multilevel Picard approximation for semilinear PIDEs and they provide a complexity analysis for their algorithm (again in the case where f does not depend on D x u).…”