  This paper contributes to the study of a new and remarkable family of
stochastic processes that we will term class $\Sigma^{r}(H)$. This class is
potentially interesting because it unifies the study of two known classes: the
class $(\Sigma)$ and the class $\mathcal{M}(H)$. In other words, we consider
the stochastic processes $X$ which decompose as $X=m+v+A$, where $m$ is a local
martingale, $v$ and $A$ are finite variation processes such that $dA$ is
carried by $\{t\geq0:X_{t}=0\}$ and the support of $dv$ is $H$, the set of
zeros of some continuous martingale $D$. First, we introduce a general
framework. Thus, we provide some examples of elements of the new class and
present some properties. Second, we provide a series of characterization
results. Afterwards, we derive some representation results which permit to
recover a process of the class $\Sigma^{r}(H)$ from its final value and of the
honest times $g=\sup\{t\geq0:X_{t}=0\}$ and $\gamma=\sup{H}$. In final, we
investigate an interesting application with processes presently studied. More
precisely, we construct solutions for skew Brownian motion equations using
stochastic processes of the class $\Sigma^{r}(H)$.
