This paper contributes to the study of class $(Σ^{r})$ as well as the càdlàg semi-martingales of class $(Σ)$, whose finite variational part is càdlàg instead of continuous. The two above-mentioned classes of stochastic processes are extensions of the family of càdlàg semi-martingales of class $(Σ)$ considered by Nikeghbali \cite{nik} and Cheridito et al. \cite{pat}; i.e., they are processes of the class $(Σ)$, whose finite variational part is continuous. The two main contributions of this paper are as follows. First, we present a new characterization result for the stochastic processes of class $(Σ^{r})$. More precisely, we extend a known characterization result that Nikeghbali established for the non-negative sub-martingales of class $(Σ)$, whose finite variational part is continuous (see Theorem 2.4 of \cite{nik}). Second, we provide a framework for unifying the studies of classes $(Σ)$ and $(Σ^{r})$. More precisely, we define and study a new larger class that we call class $(Σ^{g})$. In particular, we establish two characterization results for the stochastic processes of the said class. The first one characterizes all the elements of class $(Σ^{g})$. Hence, we derive two corollaries based on this result, which provides new ways to characterize classes $(Σ)$ and $(Σ^{r})$. The second characterization result is, at the same time, an extension of the above mentioned characterization result for class $(Σ^{r})$ and of a known characterization result of class $(Σ)$ (see Theorem 2 of \cite{fjo}). In addition, we explore and extend the general properties obtained for classes $(Σ)$ and $(Σ^{r})$ in \cite{nik,pat,mult, Akdim}.