On the Representation Theorem of G-Expectations and Paths of G--Brownian Motion

TL;DR

This paper provides a simple proof for the representation of G-expectation using a family of probability measures and demonstrates how bounded continuous functions can be approximated by cylinder functions within this framework.

Contribution

It offers an elementary proof of the G-expectation representation theorem and introduces a method to approximate continuous functions by cylinder functions.

Findings

Established a weakly compact family of probability measures for G-expectation

Proved $C_b( ext{Omega})$ is in the $ ext{E}[| ext{·}|]$-completion of $L_{ip}( ext{Omega})$

Provided a concrete approximation method for bounded continuous functions

Abstract

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures $\{P_{\theta}:\theta \in \Theta \}$ to represent an important sublinear expectation--G-expectation $\mathbb{E}[\cdot]$. We also give a concrete approximation of a bounded continuous function $X(\omega)$ by an increasing sequence of cylinder functions $L_{ip}(\Omega)$ in order to prove that $C_{b}(\Omega)$ belongs to the $\mathbb{E}[|\cdot|]$-completion of the $L_{ip}(\Omega)$.