Weak symmetric integrals with respect to the fractional Brownian motion

Abstract

The aim of this paper is to establish the weak convergence, in the topology of the Skorohod space, of the $\nu$-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value $H=(4\ell+2)^{-1}$, where $\ell=\ell(\nu)\geq1$ is the largest natural number satisfying $\int_{0}^{1}\alpha^{2j}\nu(d\alpha)=\frac{1}{2j+1}$ for all $j=0,\ldots,\ell-1$. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.