Example of a Dirichlet process whose zero energy part has finite p-th variation

Abstract

Let BH be a fractional Brownian motion on R with Hurst parameter H∈(0,1) and let F be its pathwise antiderivative (so F is a differentiable random function such that F′(x)=BxH) with F(0)=0. Let B be a standard Brownian motion, independent of BH. We show that the zero energy part At=F(Bt)−∫0tF′(Bs)dBs of F(B) has positive and finite p-th variation in a special sense for p0=2 1+H. We also present some simulation results about the zero energy part of a certain median process which suggest that its 4∕3-th variation is positive and finite.