Parameter estimation for the complex fractional Ornstein-Uhlenbeck processes with Hurst parameter H ∈ (0,1/2)

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 is an element of (1/4,1/2) and the results of Hu et al. (2019) to a two-dimensional case. When H is an element of (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 is an element of (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 is an element of (0,1/2).