Multiple fractional integral with Hurst parameter less than 1/2

Xavier Bardina and Maria Jolis

Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona
08193 Bellaterra, Barcelona, Spain

Abstract

We construct a multiple Stratonovich-type integral with respect to the fractional Brownian motion with Hurst parameter $H<\frac12$. This integral is obtained by a limit of Riemann sums procedure in the Sol\'{e} and Utzet (1990) sense. We also define the suitable traces to obtain the Hu-Meyer formula that gives the Stratonovich integral as a sum of It\^{o} integrals of these traces.
Our approach is intrinsic in the sense that we do not make use of the integral representation of the fractional Brownian motion in terms of the ordinary Brownian motion.

Keywords: fractional Brownian motion, Hu-Meyer formula, It\^{o}-type multiple integral, Stratonovich multiple integral.

Published in: Stochastic processes and their applications, 116 (3) (2006) 463-479.