Representation formulae for the fractional Brownian motion

Details Representation formulae for the fractional Brownian motion Representation formulae for the fractional Brownian motion

2009-12-16

arxiv.org/abs/0912.3168v2

Mathematics

Probability

to appear in "S\'eminaire de Probabilit\'es"

Scientific paper

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.

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