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

TL;DR

This paper derives an asymptotic expansion for the norm of a specific kernel in the Hilbert space of fractional Brownian motion, revealing conditions for oblique asymptotes and providing bounds relevant to fractional Ornstein-Uhlenbeck processes.

Contribution

It provides the first detailed asymptotic expansion of the norm of a kernel involving exponential decay in the fractional Brownian motion Hilbert space, including applications to asymptotic behavior and bounds.

Findings

Asymptotic expansion of the kernel norm up to order T^{4H-4}.

Oblique asymptote exists if and only if H ≤ 1/2.

Sharp upper bounds for the difference involving the norm and its asymptote slope.

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 $f_T(s,t):=e^{-|t-s|}\mathbf{1}_{\{0\le s,t\le T\}}$ up to the term $T^{4H-4}$. As applications, we show that the existence of the oblique asymptote of the norm $\frac12\|f_T\|^2_{{\mathfrak{H}}^{\otimes2}}$ if and only if $H\in (0,\frac12]$ and that we obtain a sharp upper bound of the difference $\left|\frac{1}{2 {T}} \|f_T\|_{\mathfrak{H}^{\otimes 2}}^2-\sigma^2\right|$ for $H\in (0,\frac34)$ which implies two significant estimates concerning to an ergodic fractional Ornstein-Uhlenbeck process, where $\sigma^2$ is the slope of the oblique asymptote.