On the existence of a time inhomogeneous skew Brownian motion and some related laws

This article is devoted to the construction of a solution for the "skew inhomogeneous Brownian motion" equation: B-t(beta) = x + W-t + integral(t)(0) beta(s)dL(s)(0)(B-beta), t >= 0. Here beta : R+ -> [-1, 1] is a Borel function, W is a standard Brownian motion, and L-t(0) (B-beta) stands for the symmetric local time at 0 of the unknown process B-beta. Using the description of the straddling excursion above a deterministic time t, we also compute the joint law of (B-t(beta), L-t(0), (B-beta), G(t)(beta)) where G(t)(beta) is the last passage time at 0 before t of B-beta.