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

We consider a multidimensional It\^o process $Y=(Y_t)_{t\in[0,T]}$ with some unknown drift coefficient process $b_t$ and volatility coefficient $\sigma(X_t,\theta)$ with covariate process $X=(X_t)_{t\in[0,T]}$, the function $\sigma(x,\theta)$ being known up to $\theta\in\Theta$. For this model we consider a change point problem for the parameter $\theta$ in the volatility component. The change is supposed to occur at some point $t^*\in (0,T)$. 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 thereoms of aymptotically mixed type.