A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4

TL;DR

This paper establishes a change of variable formula for 2D fractional Brownian motion with Hurst index H ≥ 1/4, extending known results for H > 1/4 to the critical case H=1/4 with an additional Wiener integral term.

Contribution

It introduces a novel change of variable formula for 2D fractional Brownian motion at the critical H=1/4 case, including an extra Wiener integral term.

Findings

For H > 1/4, the formula matches rough paths theory results.

At H=1/4, an additional Wiener integral term appears.

The formula extends stochastic calculus to critical fractional Brownian motion.

Abstract

We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger of equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H=1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion.