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This paper consists of two independent parts. In the first one, we contributeto the study of the class $(\Sigma)$. For instance, we provide a new way tocharacterize stochastic processes of this class. We also present some newproperties and solve the Bachelier equation. In the second part, we study theclass of stochastic processes $\Sigma(H)$. This class was introduced in\cite{f} where from tools of the theory of martingales with respect to a signedmeasure of \cite{chav}, the authors provide a general framework and methods fordealing with processes of this class. In this work, after developing some newproperties, 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 thelaw of $X_{T}$ is $\nu$.

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