A stochastic calculus for multidimensional fractional Brownian motion with arbitrary Hurst index
Résumé
We construct in this article an explicit rough path over a multi-dimensional fractional Brownian motion $B$ with arbitrary Hurst index $H$ (in particular, for $H<1/4$) by regularizing an associated random Fourier series 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 the Hopf algebra structure of the algebra of decorated rooted trees. Rough path theory gives then a general procedure to define a stochastic calculus and solve stochastic differential equations driven by this very irregular process. A variant of our regularization scheme is also expected to apply to arbitrary deterministic Hölder paths. The last section is also dedicated to the definition of a related two-dimensional Gaussian process, called {\em antisymmetric two-dimensional fractional Brownian motion}, with the same regularity as $B$ but with dependent components, to which the above construction extends naturally.
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