This paper contributes to the study of a new and remarkable family of
stochastic processes that we will term class Σr(H). This class is
potentially interesting because it unifies the study of two known classes: the
class (Σ) and the class 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≥0:Xt=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 Σr(H) from its final value and of the
honest times g=sup{t≥0:Xt=0} and γ=supH. 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 Σr(H).Comment: 23 page