Least squares estimator for Ornstein–Uhlenbeck processes driven by small fractional Lévy noises

Abstract

In this paper, we study the problem of parameter estimation for the Ornstein–Uhlenbeck processes

$\{\begin{array}{c}d{X}_t=\theta X_{t}dt+d{Y}_t\hspace{1em}\hfill \\ d{Y}_t=Y_{t}dt+\epsilon d{L}_t^d\hspace{1em}\hfill \end{array}$

driven by Ornstein–Uhlenbeck processes with small fractional Lévy noises and Y_t can be observed, based on discrete high frequency observations at regularly spaced time points $\{t_k=\frac{k}{n}$, $k=1,\dots ,n\}$ on $[0,1].$ We obtain the consistency as well as the asymptotic behavior of the least squares estimator of the unknown parameter θ when $\epsilon \to 0$ and $n\to \infty$ simultaneously.

Keywords:

• Ornstein–Uhlenbeck process
• fraction Lévy process
• least squares estimator
• consistency
• asymptotic behavior

MATHEMATICS SUBJECT CLASSIFICATION 2000:

• 60G18
• 60G22
• 65C30
• 93E24

Acknowledgments

The authors are very grateful to the anonymous referee and the editor for their insightful and valuable comments, which have improved the presentation of the paper.