The rough path associated to the multidimensional analytic fBm with any Hurst parameter

• Autores: Samy Tindel , Jérémie Unterberger
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 62, Fasc. 2, 2011, págs. 197-223
• Idioma: inglés
• DOI: 10.1007/s13348-010-0021-9
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• Resumen

  • In this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion and denoted by Γ. This process has been introduced in Unterberger (Ann Probab 37:565–614, 2009), and both its real and imaginary parts, restricted to the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on Γ can be constructed for any value of the Hurst parameter in (0,1/2). We also show how to solve differential equations driven by Γ in a neighborhood of 0 of the complex upper half-plane, by means of elementary arguments.