2025/11/07 by Chen, Yong
#41A60 #60G22 #60H07 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2511.05087
Using the inner product formula of the canonical Hilbert space of fractional Brownian motion on an interval [0,T] with Hurst parameter H∈ (0,1) given by Alazemi et al., we show the asymptotic expansion of the norm of fT(s,t):=e-|t-s|1_\0≤ s,t≤ T\ up to the term T4H-4. As applications, we show that the existence of the oblique asymptote of the norm \frac12‖fT‖2_\mathfrakH⊗2 if and only if H∈ (0,\frac12] and that we obtain a sharp upper bound of the difference |\frac12 T ‖fT‖_\mathfrakH⊗ 22-σ2| for H∈ (0,\frac34) which implies two significant estimates concerning to an ergodic fractional Ornstein-Uhlenbeck process, where σ2 is the slope of the oblique asymptote.