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

AbstractWe give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ: θ ∈ gJ} representing an important sublinear expectation— G-expectation $$ \mathbb{E} $$[·]. We also give a concrete approximation of a bounded continuous function X(ω) by an increasing sequence of cylinder functions Lip(Ω) in order to prove that Cb(Ω) belongs to the completion of Lip(Ω) under the natural norm $$ \mathbb{E} $$[| · |].