Abstract:

Our aim is to improve Hölder continuity results for the bifractional Brownian motion (bBm) \((B^{\alpha,\beta}(t))_{t\in[0,1] }\) with \(0<\alpha<1\) and \(0<\beta\leq 1\). We prove that almost all paths of the bBm belong to (resp. do not belong to) the Besov spaces \(\mathbf{Bes}(\alpha \beta,p)\) (resp. \(\mathbf{bes}(\alpha \beta,p)\)) for any \(\frac{1}{\alpha \beta}<p<\infty\), where \(\mathbf{bes}(\alpha \beta,p)\) is a separable subspace of \(\mathbf{Bes}(\alpha \beta,p)\). We also show similar regularity results in the Besov-Orlicz space \(\mathbf{Bes}(\alpha \beta, M_2)\) with \(M_2(x)=e^{x^2}-1\). We conclude by proving the Ito-Nisio theorem for the bBm with \(\alpha \beta>1/2\) in the Hölder spaces \(\mathcal{C}^{\gamma}\) with \(\gamma<\alpha \beta\).

2010 AMS Mathematics Subject Classification: Primary 60G15; Secondary 60G18, 60G17.

Keywords and phrases: bifractional Brownian motion, self-similar, Besov spaces, Besov--Orlicz spaces, Itô--Nisio.