On the Besov regularity of the bifractional Brownian motion

TL;DR

This paper enhances the understanding of the regularity of bifractional Brownian motion by establishing its Besov space properties and proving an Itô-Nisio theorem for certain parameter ranges, advancing stochastic process theory.

Contribution

The paper provides new regularity results for bifractional Brownian motion in Besov spaces and proves an Itô-Nisio theorem for it when the product of parameters exceeds 1/2.

Findings

Paths belong to Besov spaces with specific parameters.

Almost all paths do not belong to certain Besov subspaces.

Itô-Nisio theorem holds for eta > 1/2 in Hölder spaces.

Abstract

Our aim in this paper is to improve H\"{o}lder 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 (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 the It\^{o}-Nisio theorem for the bBm with $\alpha \beta>\frac{1}{2}$ in the H\"{o}lder spaces $\mathcal{C}^{\gamma}$, with $\gamma<\alpha \beta$.