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

We consider a multidimensional Itô process Y=(Y t) t∈[0,T] with some unknown drift coefficient process b t and volatility coefficient σ(X t,θ) with covariate process X=(X t) 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 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