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

TL;DR

This paper investigates the statistical inference for a complex alpha-fractional Brownian bridge, establishing well-posedness and asymptotic properties of estimators using advanced stochastic calculus techniques.

Contribution

It introduces the first analysis of the complex fractional Brownian bridge, proving estimator consistency and asymptotic distribution for all H in (0,1), with novel non-Cauchy marginals.

Findings

Proved well-posedness of the complex fractional Brownian bridge.

Established strong consistency of the least squares estimator.

Derived the asymptotic distribution with non-Cauchy marginals.

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.