Parameter Estimation for Complex α-Fractional Brownian Bridge

Abstract

We study the statistical inference problem for a complex $\hat{I}\pm$-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 $\hat{I}\pm =\hat{I}»-\sqrt{-1}w$, $\hat{I}»>0$, $w\in \mathbb{R}$ and $\hat{I}{\P }_t$ 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.

27 pages