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

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 Itô formula) has the same form as if fractional Brownian motion was 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 analogy to fractional Brownian motion fails.