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This article is devoted to the construction of a solution for the "skewinhomogeneous Brownian motion" equation, which first appear in a seminal paperby Sophie Weinryb (1983). We investigate some laws related to the constructedprocess. In particular, using the description of the straddling excursion abovea deterministic time, we compute the joint law of the process, its local timeand its straddling time.

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