A Complete Representation Theorem for $G$-martingales

TL;DR

This paper proves a comprehensive representation theorem for G-martingales, establishing existence and uniqueness of the second order term with a novel norm, advancing the mathematical understanding of stochastic processes under G-expectation.

Contribution

It introduces a complete representation theorem for G-martingales with a new norm for the second order term, ensuring existence and uniqueness, unlike previous results.

Findings

Established a complete representation theorem for G-martingales.

Proved existence and uniqueness of the second order term.

Introduced a new norm based on Song’s operator for the second order term.

Abstract

In this paper we establish a complete representation theorem for $G$-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second order term, which corresponds to the second order derivative in Markovian case. The main ingredient of the paper is a new norm for that second order term, which is based on an operator introduced by Song [26].