  We study the asymptotic behavior of the maximum likelihood estimator
corresponding to the observation of a trajectory of a Skew Brownian motion,
through a uniform time discretization. We characterize the speed of convergence
and the limiting distribution when the step size goes to zero, which in this
case are non-classical, under the null hypothesis of the Skew Brownian motion
being an usual Brownian motion. This allows to design a test on the skewness
parameter. We show that numerical simulations that can be easily performed to
estimate the skewness parameter, and provide an application in Biology.
