Some properties on G-evaluation and its applications to G-martingale decomposition

In this article, a sublinear expectation induced by G-expectation is introduced, which is called G-evaluation for convenience. As an application, we prove that for any ζ ∈ LGβ (ΩT) with some β > 1 the martingale decomposition theorem under G-expectaion holds, and that any β > 1 integrable symmetric G-martingale can be represented as an Itô integral w.r.t. G-Brownian motion. As a byproduct, we prove a regularity property for G-martingales: Any G-martingale {Mt} has a quasi-continuous version.