Least squares volatility change point estimation for partially observed diffusion processes

A one-dimensional diffusion process X = {X-t, 0 <= t <= T}, with drift b(x) and diffusion coefficient sigma(theta,x) = root theta sigma(X) known up to theta > 0, is supposed to switch volatility regime at some point t* epsilon (0,T). On the basis of discrete time observations from X, the problem is the one of estimating the instant of change in the volatility structure t* as well as the two values of , say 1 and 2, before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length Delta(n) with n Delta(n) = T. To work out our statistical problem we use a least squares approach. Consistency, rates of convergence and distributional results of the estimators are presented under an high frequency scheme. We also study the case of a diffusion process with unknown drift and unknown volatility but constant.