A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index

Jeremie Unterberger (1)

1 IECN - Institut Élie Cartan de Nancy

The main tool for stochastic calculus with respect to a multidimensional process $B$ with small Hölder regularity index is rough path theory. Once $B$ has been lifted to a rough path, a stochastic calculus -- as well as solutions to stochastic differential equations driven by $B$ -- follow by standard arguments. Although such a lift has been proved to exist by abstract arguments \cite{LyoVic07}, a first general, explicit construction has been proposed in \cite{Unt09,Unt09bis} under the name of Fourier normal ordering. The purpose of this short note is to convey the main ideas of the Fourier normal ordering method in the particular case of the iterated integrals of lowest order of fractional Brownian motion with arbitrary Hurst index.

Mots clés: fractional Brownian motion; stochastic integras; rough paths; Hopf algebra of decorated rooted trees

Domaines: Probabilités [math.PR]

Jeremie Unterberger. A Lévy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index. 2009. ⟨hal-00392336⟩