CLT for an iterated integral with respect to fBm with H > 1/2

TL;DR

This paper constructs a new type of iterated stochastic integral with fractional Brownian motion for H > 1/2, showing its equivalence to the Malliavin divergence and demonstrating convergence to Brownian motion, with applications to windings of planar fBm.

Contribution

It introduces a symmetric iterated integral with fBm, proves its equivalence to the Malliavin divergence, and establishes a convergence result to Brownian motion.

Findings

The symmetric integral equals the Malliavin divergence integral.

A family of these integrals converges in distribution to scaled Brownian motion.

Application to windings of planar fractional Brownian motion.

Abstract

We construct an iterated stochastic integral with fractional Brownian motion with H > 1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric stochastic integral is equal to the Malliavin divergence integral. By a version of the Fourth Moment theorem of Nualart and Peccati, we show that a family of such integrals converges in distribution to a scaled Brownian motion. An application is an approximation to the windings for a planar fBm, previously studied by Baudoin and Nualart.