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.