Statistical estimation of the Oscillating Brownian Motion

The Oscillating Brownian Motion is a classical, simple example of  stochastic differential equation with discontinuous diffusion coefficient. It behaves like a Brownian motion  which changes variance parameter each time it crosses a certain threshold. We consider here the problem of estimating the parameters of such process from discrete observations. Using some techniques based on approximations of quadratic variation and local time, we propose an estimator for which we prove consistence and a central limit theorem giving the rate of convergence. We consider some application to volatility modeling, being the Oscillating Brownian Motion a simple way to account of volatility clustering and leverage effect. We compare our empirical results with other regime switching models. 
(Joint work with Antoine Lejay)