Rates of convergence to the local time of Oscillating and Skew Brownian Motions

In this paper, a class of statistics based on high frequency observations of oscillating and skew Brownian motions is considered. Their convergence rate towards the local time of the underlying process is obtained in form of a Central Limit Theorem. Oscillating and skew Brownian motions are solutions to stochastic differential equations with singular coefficients: piecewise constant diffusion coefficient or drift involving the local time. The result is applied to provide estimators of the parameter of skew Brownian motion and study their asymptotic behavior. Moreover, in the case of the classical statistic given by the normalized number of crossings, the result is proved to hold for a larger class of Itô processes with singular coefficients