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This paper contributes to the study of stochastic processes of the class $(\Sigma)$. First, we extend the notion of the above-mentioned class to cadlag semi-martingales, whose finite variational part is considered cadlag instead of continuous. Thus, we present some properties and propose a method to characterize such stochastic processes. Second, we investigate continuous processes of the class $(\Sigma)$. More precisely, we derive a series of new characterization results. In addition, we construct solutions for skew Brownian motion equations using continuous stochastic processes of the class $(\Sigma)$.

An Ideal Class to Construct Solutions for Skew Brownian Motion Equations