2009/01/29 by Jérémie Unterberger, Unterberger, Jeremie · 1 citation
Mathematics · Computer Science · #Advanced Topics in Algebra #Matrix Theory and Algorithms #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.0901.4771
Fourier normal ordering \citeUnt09bis is a new algorithm to construct explicit rough paths over arbitrary Hölder-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 α (in particular, for α≤ 1/4, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of B defined in \citeUnt08. 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 \citeConKre98 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 \citeUnt09ter gives a quick overview of the method.