  This paper contributes to the study of relative martingales. Specifically,
for a closed random set $H$, they are processes null on $H$ which decompose as
$M=m+v$, where $m$ is a c\`adl\`ag uniformly integrable martingale and, $v$ is
a continuous process with integrable variations such that $v_{0}=0$ and $dv$ is
carried by $H$. First, we extend this notion to stochastic processes not
necessarily null on $H$, where $m$ is considered local martingale instead of a
uniformly integrable martingale. Thus, we provide a general framework for the
new larger class of relative martingales by presenting some structural
properties. Second, as applications, we construct solutions for skew Brownian
motion equations using continuous stochastic processes of the above mentioned
new class. In addition, we investigate stochastic differential equations driven
by a relative martingale.

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