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If we compose a smooth function g with fractional Brownian motion B withHurst 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 acontinuous function with bounded variation. In this note we prove a newintegral representation formula for the running maximum of a continuousfunction with bounded variation. Moreover we show that the analogue tofractional Brownian motion fails.

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