Parameter Estimation for Complex {\alpha}-Fractional Brownian Bridge

Abstract

We study the statistical inference problem for a complex $\alpha$-fractional Brownian bridge process $Z$ defined by the stochastic differential equation \[ \mathrm{d}Z_t = -\alpha \frac{Z_t}{T - t} \mathrm{d}t + \mathrm{d}\zeta_t, \quad t \in [0, T), \] 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 \(H \in \left(\frac{1}{2}, 1\right)\). The proofs are based on stochastic analysis elements about complex multiple Wiener-It\^o 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.