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A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering. (English) Zbl 1221.05062
Summary: Fourier normal ordering is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal 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 \(\alpha \leq 1/4\), which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of \(B\) defined by the author in [“Stochastic calculus for fractional Brownian motion with Hurst exponent \(H>\frac{1}{4}\): A rough path method by analytic extension, ”Ann. Probab. 37, No. 2, 565–614 (2009; Zbl 1172.60007)].
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 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 solving stochastic differential equations driven by \(B\). The article of the author in [“A Levy area by Fourier normal ordering for multidimensional fractional Brownian motionwith small Hurst index ”, Preprint arXiv:0906.1416v1 (2009)] gives a quick overview of the method.

MSC:
05C05 Trees
16T05 Hopf algebras and their applications
60F05 Central limit and other weak theorems
60G15 Gaussian processes
60G18 Self-similar stochastic processes
60H05 Stochastic integrals
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