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

TL;DR

This paper investigates the finite p-variation of the zero energy part of a process involving fractional Brownian motion and Brownian motion, providing theoretical results and simulation evidence for specific cases.

Contribution

It establishes that the zero energy part of a fractional Brownian motion-related process has finite p-variation for a specific p, and offers simulation insights for related median processes.

Findings

Zero energy part has positive finite p-variation for p=2/(1+H)

Simulation suggests finite 4/3-variation for median process

Theoretical and numerical evidence supports finite p-variation properties

Abstract

Let $B^H$ be a fractional Brownian motion on $\mathbb{R}$ with Hurst parameter $H\in(0,1)$, $F$ be its pathwise antiderivative with $F(0)=0$, and let $B$ be a standard Brownian motion, independent of $B^H$. We show that the zero energy part $A_t=F(B_t)-\int_0^t F'(B_s)dB_s$ of $F(B)$ has positive and finite $p$-variation in a special sense for $p_0=\frac{2}{1+H}$. We also present some simulation results about the zero energy part of a certain median process which suggest that its $4/3$-variation is positive and finite.