Supermartingale Decomposition Theorem under G-expectation

The objective of this paper is to establish the decomposition theorem for supermartingales under the $G$-framework. We first introduce a $g$-nonlinear expectation via a kind of $G$-BSDE and the associated supermartingales. We have shown that this kind of supermartingales have the decomposition similar to the classical case. The main ideas are to apply the uniformly continuous property of $S_G^beta(0,T)$, the representation of the solution to $G$-BSDE and the approximation method via penalization.