# Supermartingale Decomposition Theorem under G-expectation

## TL;DR

This paper proves a supermartingale decomposition theorem within the G-expectation framework, extending classical results to a nonlinear setting using G-BSDEs and approximation techniques.

## Contribution

It establishes a supermartingale decomposition theorem under G-expectation, introducing a new approach via G-BSDEs and approximation methods.

## Key findings

- Supermartingales under G-expectation can be decomposed similarly to classical cases.
- The paper introduces a G-nonlinear expectation using G-BSDEs.
- The decomposition is achieved through approximation and representation techniques.

## Abstract

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.