In this paper, we contribute to the study of the class $(Σ)$. In the first part of the paper, we provide new ways to characterize stochastic processes of the above mentioned class and we derive some new properties. For instance, we prove that a stochastic process $X$ is an element of the class $(Σ)$ if, and only if, its absolute value is equal to absolute value of some martingale $M$. In the second part, we study in particular, stochastic processes of the class $(Σ)$ which vanish on the zero set of a given Brownian motion. More precisely, we provide a characterization theorem and methods dealing with such stochastic processes.