Representation Formulae for the Fractional Brownian Motion

  • Jean Picard
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)


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.

Fractional Brownian motion Cameron-Martin space Laws of Gaussian processes 


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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesClermont Université, Université Blaise Pascal and CNRS UMR 6620Clermont-FerrandFrance

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