Stochastic integrals and BDG's inequalities in Orlicz-type spaces

TL;DR

This paper extends classical inequalities to stochastic integrals in Orlicz-type spaces, establishing Burkholder-Davies-Gundy inequalities for processes valued in these quasi-Banach spaces.

Contribution

It generalizes Lenglart, Lépingle, and Pratelli's inequality to continuous adapted processes in topological spaces, enabling BDG inequalities in Orlicz-type spaces.

Findings

Established BDG inequalities in Orlicz-type spaces.

Extended Lenglart-Lépingle-Pratelli inequality to topological space-valued processes.

Demonstrated applicability to cylindrical Brownian motions.

Abstract

In this paper we extend an inequality of Lenglart, L\'epingle and Pratelli \cite[Lemma 1.1]{LLP} to general continuous adapted stochastic processes with values in topology spaces. By this inequality we show Burkholder-Davies-Gundy's inequality for stochastic integrals in Orlicz-type spaces (a class of quasi-Banach spaces) with respect to cylindrical Brownian motions.