Weak symmetric integrals with respect to the fractional Brownian motion

TL;DR

This paper investigates the weak convergence of symmetric Riemann sums for fractional Brownian motion at critical Hurst parameters, leading to a distributional change-of-variable formula involving an independent Brownian motion.

Contribution

It establishes weak convergence results for symmetric integrals of fractional Brownian motion at critical Hurst values, and derives a new change-of-variable formula with an independent Brownian correction.

Findings

Weak convergence of symmetric Riemann sums at critical Hurst parameters.

A distributional change-of-variable formula involving an independent Brownian motion.

Identification of the correction term as a stochastic integral with respect to 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 $\ell=\ell(\nu)\geq 1$ is the largest natural number satisfying $\int_0^1 \alpha^{2j}\nu(d\alpha)=(2j+1)^{-1}$ 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.