Brownian semistationary processes and related processes

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