The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

TL;DR

This paper investigates the convergence properties of the Stratonovich integral for fractional Brownian motion with Hurst parameter 1/6, showing convergence in law and establishing a change of variable formula involving an independent Brownian motion.

Contribution

It demonstrates that symmetric Riemann sums for the integral converge in law and derives a change of variable formula with an Itô correction term for H=1/6.

Findings

Convergence in law of Riemann sums in Skorohod space.

Existence of a change of variable formula with correction.

Identification of an independent Brownian motion in the correction term.

Abstract

Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of c\`adl\`ag functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary It\^o integral with respect to a Brownian motion that is independent of $B$.