When does fractional Brownian motion not behave as a continuous function with bounded variation?

TL;DR

This paper investigates the behavior of fractional Brownian motion when composed with smooth functions, revealing that it does not behave as a bounded variation function, unlike continuous functions with bounded variation.

Contribution

The authors prove a new integral representation for the running maximum of bounded variation functions and demonstrate that fractional Brownian motion does not share this property.

Findings

Integral representation for the running maximum of bounded variation functions

Fractional Brownian motion does not behave as a bounded variation function

Itô formula analogy fails for fractional Brownian motion with H > 1/2

Abstract

If we compose a smooth function g with fractional Brownian motion B with Hurst index H > 1/2, then the resulting change of variables formula [or It/^o- formula] has the same form as if fractional Brownian motion would be a continuous function with bounded variation. In this note we prove a new integral representation formula for the running maximum of a continuous function with bounded variation. Moreover we show that the analogue to fractional Brownian motion fails.