A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theorem

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.

33 pages

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• A New Central Limit Theorem under Sublinear Expectations
• Sublinear Expectations and Martingales in discrete time
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• Stein's Method for Law of Large Numbers under Sublinear Expectations
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Cited by in corpus (4)

• On the rate of convergence for an $\alpha$-stable central limit theorem under sublinear expectation
• Discrete-time approximation for backward stochastic differential equations driven by $G$-Brownian motion
• G-capacity under degenerate case and its application
• A hypothesis-testing perspective on the G-normal distribution theory