Is a Brownian skew?

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.