Parameter least-squares estimation for time-inhomogeneous Ornstein–Uhlenbeck process

Abstract

Abstract We address the least-squares estimation of the drift coefficient parameter θ = ( λ , A , B , ω p ) \theta=(\lambda,A,B,\omega_{p}) of a time-inhomogeneous Ornstein–Uhlenbeck process that is observed at high frequency, in which the discretized step size ℎ satisfies h → 0 h\to 0 . In this paper, under the conditions n ⁢ h → ∞ nh\to\infty and n ⁢ h 2 → 0 nh^{2}\to 0 , we prove the consistency and the asymptotic normality of the estimators. We obtain the convergence of the parameters at rate n ⁢ h \sqrt{nh} , except for ω p \omega_{p} at n 3 ⁢ h 3 \sqrt{n^{3}h^{3}} . To verify our theoretical findings, we do a simulation study. We then illustrate the use of the proposed model in fitting the energy use of light fixtures in one Belgium household and the stock exchange.