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

TL;DR

This paper introduces $G$-evaluation, a sublinear expectation based on $G$-expectation, and applies it to establish decomposition and representation theorems for $G$-martingales, including their quasi-continuity.

Contribution

It develops the concept of $G$-evaluation and proves new decomposition and representation results for $G$-martingales under this framework.

Findings

Decomposition theorem for $L^eta_G$-integrable variables.

Representation of symmetric $G$-martingales as Itô integrals.

Existence of quasi-continuous versions for $G$-martingales.

Abstract

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 any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ the decomposition theorem holds and any $\beta>1$ integrable symmetric $G$-martingale can be represented as an It$\hat{o}'s$ integral w.r.t $G$-Brownian motion. As a byproduct, we prove a regular property for $G$-martingale: Any $G$-martingale $\{M_t\}$ has a quasi-continuous version