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

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 $\nu$.