Parameter Estimation for the Complex Fractional Ornstein-Uhlenbeck Processes with Hurst parameter H \in (0, 1/2)

• Main Authors: Alazemi, Fares, Alsenafi, Abdulaziz, Chen, Yong, Zhou, Hongjuan
• Format: Journal Article
• Language: English
• Published: 25.06.2024
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2406.18004

We study the strong consistency and asymptotic normality of a least squares estimator of the drift coefficient in complex-valued Ornstein-Uhlenbeck processes driven by fractional Brownian motion, extending the results of Chen, Hu, Wang (2017) to the case of Hurst parameter H \in (1/4 , 1/2) and the results of Hu, Nualart, Zhou (2019) to a two-dimensional case. When H \in (0, 1/4], it is found that the integrand of the estimator is not in the domain of the standard divergence operator. To facilitate the proofs, we develop a new inner product formula for functions of bounded variation in the reproducing kernel Hilbert space of fractional Brownian motion with Hurst parameter H \in (0, 1/2). This formula is also applied to obtain the second moments of the so-called {\alpha}-order fractional Brownian motion and the {\alpha}-fractional bridges with the Hurst parameter H \in (0, 1/2).