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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.05087 |
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Table of Contents:
- 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 $\left|\frac{1}{2 {T}} \|f_T\|_{\mathfrak{H}^{\otimes 2}}^2-σ^2\right|$ for $H\in (0,\frac34)$ which implies two significant estimates concerning to an ergodic fractional Ornstein-Uhlenbeck process, where $σ^2$ is the slope of the oblique asymptote.