Least Squares Volatility Change Point Estimation for Partially Observed Diffusion Processes

Abstract

A one-dimensional diffusion process X = {X t , 0 ≤ t ≤ T}, with drift b(x) and diffusion coefficient known up to θ > 0, is supposed to switch volatility regime at some point t* ∈ (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 θ, say θ1 and θ2, before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length Δ n with nΔ 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.