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Parameter Estimation for Complex α-Fractional Brownian Bridge

Yong Chen, Lin Fang, Ying Li, Hongjuan Zhou

Abstract

We study the statistical inference problem for a complex $α$-fractional Brownian bridge process $Z$ defined by the stochastic differential equation \[ \mathrm{d}Z_t = -α\frac{Z_t}{T - t} \mathrm{d}t + \mathrm{d}ζ_t, \quad t \in [0, T), \] with initial condition $Z_0 = 0$, where $α= λ- \sqrt{-1}w$, $λ> 0$, $w \in \mathbb{R}$ and $ζ_t$ is a complex fractional Brownian motion. We establish the well-posedness of the fractional Brownian bridge $Z_t$ over the time interval $[0, T]$ for all $H \in (0, 1)$, and prove the strong consistency and the asymptotic distribution for the classic least squares estimator of the parameter \(α\) when \(H \in \left(\frac{1}{2}, 1\right)\). The proofs are based on stochastic analysis elements about complex multiple Wiener-Itô integrals and the complex Malliavin calculus. Unlike the real-valued fractional Brownian bridge considered in the literature, the two-dimensional limiting distribution has non-Cauchy marginal distributions.

Parameter Estimation for Complex α-Fractional Brownian Bridge

Abstract

We study the statistical inference problem for a complex -fractional Brownian bridge process defined by the stochastic differential equation with initial condition , where , , and is a complex fractional Brownian motion. We establish the well-posedness of the fractional Brownian bridge over the time interval for all , and prove the strong consistency and the asymptotic distribution for the classic least squares estimator of the parameter when \(H \in \left(\frac{1}{2}, 1\right)\). The proofs are based on stochastic analysis elements about complex multiple Wiener-Itô integrals and the complex Malliavin calculus. Unlike the real-valued fractional Brownian bridge considered in the literature, the two-dimensional limiting distribution has non-Cauchy marginal distributions.
Paper Structure (8 sections, 16 theorems, 115 equations)

This paper contains 8 sections, 16 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Complex Wiener-Itô multiple integrals
  4. Complex Malliavin calculus
  5. Results regarding a real-valued fractional Brownian bridge
  6. Existence of complex $\alpha$-fractional Brownian bridge
  7. The strong consistency of the LSE
  8. The asymptotic distribution

Key Result

Theorem 1.1

Assume $H \in (0, 1)$. Suppose $\lambda = \mathop{\mathrm{Re}}\nolimits(\alpha) \in (0, H)$. Then the limit $\omega_T:=\lim_{t \uparrow T} \omega_t$ exists in $L^2$ and almost surely. Moreover, Consequently, the Gaussian process $\omega:=\{\omega_t, t \in [0, T]\}$ admits a modification on $[0,T]$ with $(H-\lambda-\epsilon)$-Hölder continuous paths for all $\epsilon\in (0,H-\lambda)$. In addition

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 16 more