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

We address the least-squares estimation of the drift coefficient parameter θ=(λ,A,B,ωp)\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→0h\to 0. In this paper, under the conditions n⁢h→∞nh\to\infty and n⁢h2→0nh^{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 ωp\omega_{p} at n3⁢h3\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.