Probabilistic approximation of fully nonlinear second-order PIDEs with convergence rates for the universal robust limit theorem

TL;DR

This paper introduces a probabilistic approximation scheme for complex nonlinear PIDEs linked to Levy processes, providing explicit error bounds and convergence rates for robust limit theorems.

Contribution

It develops a recursive approximation method for nonlinear PIDEs with explicit error estimates, advancing numerical analysis under sublinear expectations.

Findings

Established explicit error bounds for the approximation scheme.

Proved convergence rates for the universal robust limit theorem.

Unified treatment of several classical limit theorems with quantitative bounds.

Abstract

This paper develops a probabilistic approximation scheme for a class of nonstandard, fully nonlinear second-order partial integro-differential equations (PIDEs) associated with nonlinear Levy processes under Peng's G-expectation framework. The PIDE features a supremum over a family of alpha-stable Levy measures, possibly degenerate diffusion coefficients, and a non-separable uncertainty set, which places it outside the scope of existing numerical theories for PIDEs. We construct a recursive, piecewise-constant approximation of the viscosity solution and establish explicit error estimates for the scheme. As a key application, our results yield quantitative convergence rates for the universal robust limit theorem under sublinear expectations. This provides a unified treatment of Peng's robust central limit theorem and law of large numbers, as well as the alpha-stable limit theorem of Bayraktar and Munk, together with explicit Berry-Esseen-type bounds.