A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theorem
Shuo Huang, Gechun Liang
TL;DR
This work develops a monotone, finite-Δ approximation scheme for the G-equation under G-expectations, establishing convergence to the viscosity solution with explicit error bounds. By linking PDE numerical analysis with probabilistic central limit phenomena, it provides the first explicit Berry-Esseen-type convergence rate for Peng's robust central limit theorem and an associated convergence framework for the Black-Scholes-Barenblatt equation. The approach yields sharp, computable constants and extends to both degenerate and non-degenerate regimes, offering insights into the interplay between monotone schemes for viscosity solutions and robust probabilistic limits with model uncertainty. Practical implications include accurate numerical schemes for G-expectations, robust option pricing under volatility ambiguity, and improved quantitative understanding of convergence rates in nonlinear probabilistic limit theorems.
Abstract
We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations. Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space. The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation. We establish the convergence of the scheme and determine the convergence rate with an explicit error bound, using the comparison principles for both the scheme and the equation together with a mollification procedure. The first application is obtaining the convergence rate of Peng's robust central limit theorem with an explicit bound of Berry-Esseen type. The second application is an approximation scheme with its convergence rate for the Black-Scholes-Barenblatt equation.