Characterization of a new class of stochastic processes including all known extensions of the class $(Sigma)$

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