## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 2243-2267.

Dates
Revised: July 2017
First available in Project Euclid: 13 June 2018

https://projecteuclid.org/euclid.aop/1528876827

Digital Object Identifier
doi:10.1214/17-AOP1227

Mathematical Reviews number (MathSciNet)
MR3813991

Zentralblatt MATH identifier
06919024

#### Citation

Binotto, Giulia; Nourdin, Ivan; Nualart, David. Weak symmetric integrals with respect to the fractional Brownian motion. Ann. Probab. 46 (2018), no. 4, 2243--2267. doi:10.1214/17-AOP1227. https://projecteuclid.org/euclid.aop/1528876827

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