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 s(theta, x)=sqrt(theta) s(x) known up to theta>0, is supposed to switch volatility regime at some point t* in (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 theta, say theta_1 and theta_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.