Representation formulae for the fractional Brownian motion

TL;DR

This paper reviews classical representations of fractional Brownian motion, introduces fractional calculus concepts, and compares properties of its Cameron-Martin space, providing unified proofs and new insights.

Contribution

It offers a unified treatment of existing representations, introduces fractional calculus tools, and compares laws of fractional Brownian motion variants.

Findings

Provides alternative proofs of known results

Analyzes properties of the Cameron-Martin space

Compares laws of fractional Brownian motion variants

Abstract

We discuss the relationships between some classical representations of the fractional Brownian motion, as a stochastic integral with respect to a standard Brownian motion, or as a series of functions with independent Gaussian coefficients. The basic notions of fractional calculus which are needed for the study are introduced. As an application, we also prove some properties of the Cameron-Martin space of the fractional Brownian motion, and compare its law with the law of some of its variants. Several of the results which are given here are not new; our aim is to provide a unified treatment of some previous literature, and to give alternative proofs and additional results; we also try to be as self-contained as possible.