Multiple fractional integral with Hurst parameter less than 12

We construct a multiple Stratonovich-type integral with respect to the fractional Brownian motion with Hurst parameter H<12. This integral is obtained by a limit of Riemann sums procedure in the Solé and Utzet [Stratonovich integral and trace, Stochastics Stochastics Rep. 29 (2) (1990) 203–220] sense. We also define the suitable traces to obtain the Hu–Meyer formula that gives the Stratonovich integral as a sum of Itô 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.