Least-squares estimation for the Vasicek model driven by the complex fractional Brownian motion

Abstract

In this paper, we study the least-squares estimation problem for the Vasicek model $\mathrm{d}{Z}_t=a(b-Z_t)\mathrm{d}t+\mathrm{d}{\xi }_t^H,t\ge 0,$ driven by the complex fractional Brownian motion ${\xi }_t^H=\frac{B_t^{H,1}+i{B}_t^{H,2}}{\sqrt{2}},$ where $(B_t^{H,1},B_t^{H,2})$ is a two-dimensional fractional Brownian motion with Hurst parameter $H\in (\frac{1}{2},\frac{3}{4}).$ We obtain the strong consistency and asymptotic normality of ${\hat{a}}_T$ and ${\hat{b}}_T$ using the Garsia–Rodemich–Rumsey inequality and complex fourth moment theorems.

Keywords:

• Fractional Brownian motion
• complex multiple Wiener–Itô integral
• least-squares estimator
• fourth moment theorems
• consistency
• asymptotic normality

2010 Mathematics Subject Classifications:

• 60G15
• 60G18
• 60G22
• 60H07

Acknowledgments

The authors would like to thank the anonymous referee and editor whose remarks and suggestions greatly improved the presentation of our paper. The authors also thank Professor Yong Chen for very stimulating discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [12071003, 11901005].