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

TL;DR

This paper introduces a novel Fourier normal ordering algorithm to explicitly construct rough paths over multidimensional fractional Brownian motion with any Hurst index, including challenging cases with low Hurst parameters.

Contribution

It extends the Fourier normal ordering method to fractional Brownian motion with arbitrary Hurst index, solving an open problem for indices less than or equal to 1/4.

Findings

Constructed explicit rough paths for all Hurst indices, including  

Proved the rough path has finite moments

Demonstrated the algebraic properties using Hopf algebras

Abstract

Fourier normal ordering \cite{Unt09bis} is a new algorithm to construct explicit rough paths over arbitrary H\"older-continuous multidimensional paths. We apply in this article the Fourier normal ordering ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion $B$ with arbitrary Hurst index $\alpha$ (in particular, for $\alpha\le 1/4$, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of $B$ defined in \cite{Unt08}. The regularization procedure is applied to 'Fourier normal ordered' iterated integrals obtained by permuting the order of integration so that innermost integrals have highest Fourier modes. The algebraic properties of this rough path are best understood using two Hopf algebras: the Hopf algebra of decorated rooted trees \cite{ConKre98} for the multiplicative or Chen 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 have finite moments. As well-known, the construction of a rough path is the key to defining a stochastic calculus and solve stochastic differential equations driven by $B$. The article \cite{Unt09ter} gives a quick overview of the method.