New contributions to the study of stochastic processes of the class $(\Sigma)$

TL;DR

This paper advances the understanding of stochastic processes in class $(\Sigma)$ by offering new characterizations, properties, and specific analysis of processes vanishing on Brownian motion zero sets.

Contribution

It introduces novel characterizations and properties of class $(\Sigma)$ processes and explores their behavior when vanishing on Brownian motion zero sets.

Findings

A process in class $(\Sigma)$ is equivalent to the absolute value of a martingale.

New characterization theorems for processes vanishing on Brownian zero sets.

Methods for analyzing processes that vanish on specific Brownian motion zero sets.

Abstract

In this paper, we contribute to the study of the class $(\Sigma)$. In the first part of the paper, we provide new ways to characterize stochastic processes of the above mentioned class and we derive some new properties. For instance, we prove that a stochastic process $X$ is an element of the class $(\Sigma)$ if, and only if, its absolute value is equal to absolute value of some martingale $M$. In the second part, we study in particular, stochastic processes of the class $(\Sigma)$ which vanish on the zero set of a given Brownian motion. More precisely, we provide a characterization theorem and methods dealing with such stochastic processes.