An asymptotic expansion of the norm of $e^{-|t-s|}{1}_{\{0\le s,t\le T\}}$ in the canonical Hilbert space of fractional Brownian motion

Abstract

Using the inner product formula of the canonical Hilbert space of fractional Brownian motion on an interval $[0,T]$ with Hurst parameter $H\in (0,1)$ given by Alazemi et al., we show the asymptotic expansion of the norm of $f_T(s,t):=e^{-|t-s|}{1}_{\{0\le s,t\le T\}}$ up to the term $T^{4H-4}$. As applications, we show that the existence of the oblique asymptote of the norm $\frac{1}{2}\Vert f_T{\Vert }_{{\mathfrak{H}}^{\otimes 2}}^2$ if and only if $H\in (0,\frac{1}{2}]$ and that we obtain a sharp upper bound of the difference $\left|\frac{1}{2T}\Vert f_T{\Vert }_{{\mathfrak{H}}^{\otimes 2}}^2-{\sigma }^2\right|$ for $H\in (0,\frac{3}{4})$ which implies two significant estimates concerning to an ergodic fractional Ornstein-Uhlenbeck process, where ${\sigma }^2$ is the slope of the oblique asymptote.

22 pages