Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (2): 573-595.doi: 10.1007/s10473-021-0218-x

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PARAMETER ESTIMATION FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A GENERAL GAUSSIAN NOISE

Yong CHEN1, Hongjuan ZHOU2   

  1. 1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China;
    2. School of Mathematical and Statistical Sciences, Arizona State University, Arizona 85287, USA
  • Received:2020-02-29 Revised:2020-03-08 Online:2021-04-25 Published:2021-04-29
  • Contact: Hongjuan ZHOU E-mail:Hongjuan.Zhou@asu.edu
  • About author:Yong CHEN,E-mail:zhishi@pku.org.cn,chenyong77@gmail.com
  • Supported by:
    Dr. Yong Chen is supported by NSFC (11871079, 11961033, and 11961034).

Abstract: In this paper, we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by $(ts)^{\beta-1}$ up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments; some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise $(G_t)_{t\ge 0}$. For the least squares estimator and the second moment estimator constructed from the continuous observations, we prove the strong consistency and the asympotic normality, and obtain the Berry-Esséen bounds. The proof is based on the inner product's representation of the Hilbert space $\mathfrak{H}$ associated with the Gaussian noise $(G_t)_{t\ge 0}$, and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.

Key words: Fourth moment theorem, Ornstein-Uhlenbeck process, Gaussian process, Malliavin calculus

CLC Number: 

  • 60H07
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