  Contributions of the present paper consist of two parts. In the first one, we
contribute to the theory of stochastic calculus for signed measures. For
instance, we provide some results permitting to characterize martingales and
Brownian motion both defined under a signed measure. We also prove that the
uniformly integrable martingales (defined with respect to a signed measure) can
be expressed as relative martingales and we provide some new results to the
study of the class $\Sigma(H)$ which appeared for the first time in \cite{f}
and studied in \cite{f,e,o}. The second part is devoted to the construction of
solutions for the \textbf{homogeneous skew Brownian motion equation} and for
the \textbf{inhomogeneous skew Brownian motion equation}. To do this, our
ingredients are the techniques and results developed in the first part that we
apply on some stochastic processes borrowed from the theory of stochastic
calculus for signed measures. Our methods are inspired by those used by
Bouhadou and Ouknine in \cite{siam}. Moreover, their solution of the
inhomogeneous skew Brownian motion equation is a particular case of those we
propose in this paper.
