Doob's optional sampling and maximal inequality for $G$-martingales

TL;DR

This paper extends classical martingale results to the $G$-framework, establishing a Doob's optional sampling theorem and a maximal inequality, which enhance the understanding and representation of $G$-martingales.

Contribution

It introduces a Doob's optional sampling theorem and a maximal inequality for $G$-martingales, improving existing representation theorems.

Findings

Established Doob's optional sampling in $G$-framework

Proved the $G$-martingale maximal inequality

Enhanced $G$-martingale representation theorems

Abstract

The paper considers the martingale theory in the $G$-framework. A form of Doob's optional sampling is established, which allows to prove the exact analogue of the classical maximal inequality. The obtained results are used to improve the existing $G$-martingale representation theorems.