Representation formulae for the fractional Brownian motion

By Jean Picard. 2009

61 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/b95ae6e638496b4fb20e3787b7fb0117f0cd2b15

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Citations in the corpus

Noiseless regularisation by noise (2020, background): “The material on fractional Brownian motion presented here is classical and taken from [40] and [44].”
Distribution dependent SDEs driven by additive fractional Brownian motion (2021, background): “All the results we are going to recall here are classical and can be found in [34, 37].”
An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr\"odinger equation (2017, background): “Then, Picard [30] showed the following result on the relation between BH and B̂H .”