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

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