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 $0<\beta \le 1$. 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 $\frac{1}{\alpha \beta }<p<\mathrm{\infty }$, 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.