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.
Categories: math.PR
