Least Squares Estimation for α -Fractional Bridge with Discrete Observations

Abstract

We consider a fractional bridge defined as , where $B^H$ is a fractional Brownian motion of Hurst parameter $H>\mathrm{1}/\mathrm{2}$ and parameter $\alpha >\mathrm{0}$ is unknown. We are interested in the problem of estimating the unknown parameter $\alpha >\mathrm{0}$. Assume that the process is observed at discrete time $t_i=i{\Delta }_n,\hspace{0.17em}\hspace{0.17em}i=\mathrm{0},\dots ,n$, and $T_n=n{\Delta }_n$ denotes the length of the “observation window.” We construct a least squares estimator ${\widehat{\alpha }}_n$ of $\alpha$ which is consistent; namely, ${\widehat{\alpha }}_n$ converges to $\alpha$ in probability as $n\to \mathrm{\infty }$.