Statistical inference for Vasicek-type model driven by self-similar Gaussian processes

Abstract

In this paper, we consider the drift parameters estimation problem for the Vasicek-type model defined as $$d{X}_t=a(b-X_t)dt+d{G}_t, X_0=0, t\ge 0$$ where a < 0 and $b\in \mathbb{R}$ are considered as unknown drift parameters and G_t is a self-similar Gaussian process with index $L\in (1/2,1)$. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distributions of our estimators $\widehat{a}$ of a and $\widehat{b}$ of b based on the observation ${\left\{X_t\right\}}_{t\in [0,T]}$ as $T\to \infty$. Our approach extend the result of Xiao and Yu (2017) for the case when G is a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{2},1)$. We also discuss the cases of sub-fractional Browian motion and bi-fractional Brownian motion. The conclusion can also be extended to more general self-similarity processes, such as Hermite processes.

Keywords:

• Parameter estimation
• strong consistency
• self-similarity processes
• Vasicek-type model

Mathematics Subject Classification:

• 60G18
• 65C30
• 93E24

Acknowledgement

The author is grateful to the anonymous referees and the editor for their insightful and valuable comments which have greatly improved the presentation of the paper.