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

Fares Alazemi, Abdulaziz Alsenafi, Yong Chen, Hongjuan Zhou

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 α-order fractional Brownian motion and the α-fractional bridges with the Hurst parameter H \in (0, 1/2).

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

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 α-order fractional Brownian motion and the α-fractional bridges with the Hurst parameter H \in (0, 1/2).

Paper Structure

This paper contains 7 sections, 20 theorems, 104 equations.

Table of Contents

  1. Introduction and main results
  2. Hilbert space associated with fBm and complex isonormal Gaussian process
  3. Complex Wiener-Itô multiple integrals and complex fourth moment theorem
  4. The second moments of two double Wiener-Itô Integrals
  5. The statistical inference of the two-dimensional fractional Ornstein-Ulenbeck process with Hurst parameter $H\in (0,\frac{1}{2})$
  6. The existence of the $\alpha$-fractional Brownian bridge with Hurst parameter $H\in (0,\frac{1}{2})$
  7. Appendix

Key Result

Theorem \oldthetheorem

Let $H\in (\frac{1}{4}, \frac{3}{4})$.

Theorems & Definitions (27)

  • Theorem \oldthetheorem
  • Remark 1
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark 2
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 17 more