Parameter Estimation for Complex α-Fractional Brownian Bridge

We study the statistical inference problem for a complex $\alpha$-fractional Brownian bridge process $Z$ defined by the stochastic differential equation $$\mathrm{d}{Z}_t=-\alpha \frac{Z_t}{T-t}\mathrm{d}t+\mathrm{d}{\zeta }_t,t\in [0,T),$$ with initial condition $Z_0=0$, where $\alpha =\lambda -\sqrt{-1}w$, $\lambda >0$, $w\in \mathbb{R}$ and ${\zeta }_t$ 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 $\alpha$ when $H\in (\frac{1}{2},1)$. 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.