Estimation for the change point of the volatility in a stochastic differential equation ∗

We consider a multidimensional Itô process Y = (Yt)t∈[0,T ] with some unknown drift coefficient process bt and volatility coefficient σ(Xt, θ) with covariate process X = (Xt)t∈[0,T ], the function σ(x, θ) being known up to θ ∈ Θ. For this model we consider a change point problem for the parameter θ in the volatility component. The change is supposed to occur at some point t∗ ∈ (0, T ). Given discrete time observations from the process (X,Y ), we propose quasimaximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit thereoms of aymptotically mixed type.