An Iterative Approximation of the Sublinear Expectation of an Arbitrary Function of $G$-normal Distribution and the Solution to the Corresponding $G$-heat Equation

Abstract

It has been a well-known problem in the $G$-framework that it is hard to compute the sublinear expectation of the $G$-normal distribution $\hat{\mathbb{E}}[\varphi(X)]$ when $\varphi$ is neither convex nor concave, if not involving any PDE techniques to solve the corresponding $G$-heat equation. Recently, we have established an efficient iterative method able to compute the sublinear expectation of \emph{arbitrary} functions of the $G$-normal distribution, which directly applies the \emph{Nonlinear Central Limit Theorem} in the $G$-framework to a sequence of variance-uncertain random variables following the \emph{Semi-$G$-normal Distribution}, a newly defined concept with a nice \emph{Integral Representation}, behaving like a ladder in both theory and intuition, helping us climb from the ground of classical normal distribution to approach the peak of $G$-normal distribution through the \emph{iteratively maximizing} steps. The series of iteration functions actually produce the whole \emph{solution surface} of the $G$-heat equation on a given time grid.