The Least Squares Estimator for an Ornstein-Uhlenbeck Process Driven by a Hermite Process with a Periodic Mean

Abstract

We consider the least square estimator for the parameters of Ornstein-Uhlenbeck processes $$d{Y_s} = \left( {\sum\limits_{j = 1}^k {{\mu _j}{\phi _j}\left( s \right) - \beta {Y_s}} } \right){\rm{d}}s + {\rm{dZ}}_s^{q,H},$$ driven by the Hermite process Z with order q ≥ 1 and a Hurst index H ∈ (½, 1), where the periodic functions φj(s),j = 1,⌦, k are bounded, and the real numbers μj, j = 1, …, k together with β > 0 are unknown parameters. We establish the consistency of a least squares estimation and obtain the asymptotic behavior for the estimator. We also introduce alternative estimators, which can be looked upon as an application of the least squares estimator. In terms of the fractional Ornstein-Uhlenbeck processes with periodic mean, our work can be regarded as its non-Gaussian extension.