Parameter estimation for the complex fractional Ornstein–Uhlenbeck processes with Hurst parameter H∈(0,12)

• Published in: Chaos, solitons and fractals Vol. 188; p. 115556
• Main Authors: Alazemi, Fares, Alsenafi, Abdulaziz, Chen, Yong, Zhou, Hongjuan
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.11.2024
• Subjects: Complex Wiener–Itô multiple integral; Fourth moment theorem; Fractional Brownian motion; Fractional Ornstein–Uhlenbeck process; Least squares estimate; α-fractional Brownian bridge; α-order fractional Brownian motion; 60G22; α-fractional Brownian bridge; 62M09; Least squares estimate; Complex Wiener–Itô multiple integral; Fractional Brownian motion; α-order fractional Brownian motion; 60G15; Fractional Ornstein–Uhlenbeck process; Fourth moment theorem
• DOI: 10.1016/j.chaos.2024.115556
• ISSN: 0960-0779

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 et al. (2017) to the case of Hurst parameter H∈(14,12) and the results of Hu et al. (2019) to a two-dimensional case. When H∈(0,14], 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∈(0,12). This formula is also applied to obtain the second moments of the so-called α-order fractional Brownian motion and the α-fractional bridges with the Hurst parameter H∈(0,12). •We develop a new inner product formula for functions in the reproducing kernel Hilbert space of fBm.•We prove the asymptotics for the LS estimator of the drift coefficient in complex fOU processes.•We estimate the second moments of the α-order fBm and the α-fractional bridges.