2021/11/30 by Yong Chen, Chen, Yong, Xiangmeng Gu +3
Economics, Econometrics and Finance · Mathematics · #60F25 #60H07 #62M09 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2111.15292
openalex publication_date 2021/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In Chen and Zhou 2021, they consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process (Gt)t≥ 0. The second order mixed partial derivative of the covariance function R(t, s)=𝔼[Gt Gs] can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other is bounded by (ts)H-1 with H∈ (\frac12, 1), up to a constant factor. In this paper, we investigate the same problem but with the assumption of H∈ (0, \frac12). It is well known that there is a significant difference between the Hilbert space associated with the fractional Gaussian processes in the case of H∈ (\frac12, 1) and that of H∈ (0, \frac12). The starting point of this paper is a new relationship between the inner product of \mathfrakH associated with the Gaussian process (Gt)t≥ 0 and that of the Hilbert space \mathfrakH1 associated with the fractional Brownian motion (BHt)t≥ 0. Then we prove the strong consistency with H∈ (0, \frac12), and the asymptotic normality and the Berry-Esséen bounds with H∈ (0,\frac38) for both the least squares estimator and the moment estimator of the drift parameter constructed from the continuous observations. A good many inequality estimates are involved in and we also make use of the estimation of the inner product based on the results of \mathfrakH1 in Hu, Nualart and Zhou 2019.