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

TL;DR

This paper investigates the properties of a least squares estimator for the drift in complex-valued Ornstein-Uhlenbeck processes driven by fractional Brownian motion, extending previous results to a broader Hurst parameter range and developing new mathematical tools.

Contribution

It extends the analysis of the estimator's properties to Hurst parameters in (1/4, 1/2) and introduces a new inner product formula for functions of bounded variation in the fractional Brownian motion context.

Findings

Estimator is strongly consistent for H in (1/4, 1/2)

Asymptotic normality established for the estimator

New inner product formula for functions of bounded variation

Abstract

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).