Some contributions to the study of stochastic processes of the classes $\Sigma(H)$ and $(\Sigma)$

TL;DR

This paper advances the understanding of stochastic processes in classes nd y characterizing, analyzing properties, and solving equations related to these processes, including embedding measures within them.

Contribution

It introduces new characterizations and properties of the classes nd nd develops methods for measure embedding and solving the Bachelier equation.

Findings

New characterizations of lass processes

Properties and solutions for the Bachelier equation

Embedding non-atomic measures in lass processes

Abstract

This paper consists of two independent parts. In the first one, we contribute to the study of the class $(\Sigma)$. For instance, we provide a new way to characterize stochastic processes of this class. We also present some new properties and solve the Bachelier equation. In the second part, we study the class of stochastic processes $\Sigma(H)$. This class was introduced in \cite{f} where from tools of the theory of martingales with respect to a signed measure of \cite{chav}, the authors provide a general framework and methods for dealing with processes of this class. In this work, after developing some new properties, we embed a non-atomic measure $\nu$ in $X$, a process of the class $\Sigma(H)$. More precisely, we find a stopping time $T<\infty$ such that the law of $X_{T}$ is $\nu$.