English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Fourier normal ordering \cite{Unt09bis} is a new algorithm to constructexplicit rough paths over arbitrary H\"older-continuous multidimensional paths.We apply in this article the Fourier normal ordering ordering algorithm to theconstruction of an explicit rough path over multi-dimensional fractionalBrownian motion $B$ with arbitrary Hurst index $\alpha$ (in particular, for$\alpha\le 1/4$, which was till now an open problem) by regularizing theiterated integrals of the analytic approximation of $B$ defined in\cite{Unt08}. The regularization procedure is applied to 'Fourier normalordered' iterated integrals obtained by permuting the order of integration sothat innermost integrals have highest Fourier modes. The algebraic propertiesof this rough path are best understood using two Hopf algebras: the Hopfalgebra of decorated rooted trees \cite{ConKre98} for the multiplicative orChen property, and the shuffle algebra for the geometric or shuffle property.The rough path lives in Gaussian chaos of integer orders and is shown to havefinite moments. As well-known, the construction of a rough path is the key todefining a stochastic calculus and solve stochastic differential equationsdriven by $B$. The article \cite{Unt09ter} gives a quick overview of themethod.

A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering