Brownian semistationary processes and related processes

TL;DR

This paper provides a pathwise decomposition of Brownian semistationary processes into fractional Brownian motions, enabling analysis of their properties and deriving Itô's formula for a specific subclass.

Contribution

It introduces a novel pathwise decomposition for a class of Brownian semistationary processes using fractional Brownian motions, expanding understanding of their structure and properties.

Findings

Decomposition of BSS processes into fractional Brownian motions

Path properties of the specialized BSS subclass analyzed

Itô's formula derived for the specific BSS subclass

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 $\varphi_{\alpha}\left(x\right)=L\left(x\right)x^{\alpha}$ 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 It\^o's formula for this subclass of $\mathcal{BSS}$ processes.