Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes

TL;DR

This paper explores the properties of function spaces related to G-Brownian motion paths under sublinear expectations, providing foundational results and a generalized Kolmogorov criterion with applications in finance and risk assessment.

Contribution

It introduces new properties of Banach spaces associated with G-Brownian motion and extends Kolmogorov's criterion for processes under sublinear expectations.

Findings

Established key properties of function spaces induced by G-expectation.

Developed a generalized Kolmogorov criterion for continuous modifications.

Applied results to risk measures in finance under volatility uncertainty.

Abstract

In this paper we give some basic and important properties of several typical Banach spaces of functions of $G$-Brownian motion pathes induced by a sublinear expectation--G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. The results can be applied to continuous time dynamic and coherent risk measures in finance in particular for path-dependence risky positions under situations of volatility model uncertainty.