Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion

TL;DR

This paper investigates the asymptotic convergence of weighted quadratic and cubic variations of fractional Brownian motion, revealing new conditions under which these variations converge in $L^2$ to explicit limits depending on the process.

Contribution

It provides new convergence results for weighted variations of fractional Brownian motion using Malliavin calculus, especially for Hurst indices less than 1/4 and 1/6.

Findings

Convergence in $L^2$ for quadratic variations when $H<1/4$

Convergence in $L^2$ for cubic variations when $H<1/6$

Explicit limit depends only on the fractional Brownian motion

Abstract

The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion $B$ with Hurst index $H$. In the quadratic (resp. cubic) case, when $H<1/4$ (resp. $H<1/6$), we show by means of Malliavin calculus that the convergence holds in $L^2$ toward an explicit limit which only depends on $B$. This result is somewhat surprising when compared with the celebrated Breuer and Major theorem.