Brownian semistationary processes and related processes

Abstract

In this paper we find a pathwise decomposition of a certain class of Brownian semistationary processes ($\mathcal{BSS}$) in terms of fractional Brownian motions. To do this, we specialize in the case when the kernel of the $\mathcal{BSS}$ is given by ${\phi }_{\alpha }\left(x\right)=L\left(x\right)x^{\alpha }$ with $\alpha \in (-1/2,0)\cup (0,1/2)$ and $L$ a continuous function slowly varying at zero. We use this decomposition to study some path properties and derive Itô's formula for this subclass of $\mathcal{BSS}$ processes.