On the $\Phi$-variation of stochastic processes with exponential moments

TL;DR

This paper establishes sharp conditions for the exponential integrability of stochastic processes to have bounded $ ext{ extPhi}$-variation, providing bounds for Gaussian processes and analyzing the roughness of Hermite processes like the Rosenblatt process.

Contribution

It introduces optimal $ ext{ extPhi}$-variation bounds for Hermite processes, revealing their roughness compared to fractional Brownian motion.

Findings

Hermite processes have bounded $ ext{ extPhi}$-variation with a specific optimal $ ext{ extPhi}$.

The Rosenblatt process is rougher than fractional Brownian motion in $ ext{ extPhi}$-variation terms.

Sharp sufficient conditions for exponential integrability of sample paths are derived.

Abstract

We obtain sharp sufficient conditions for exponentially integrable stochastic processes $X=\{X(t)\!\!: t\in [0,1]\}$, to have sample paths with bounded $\Phi$-variation. When $X$ is moreover Gaussian, we also provide a bound of the expectation of the associated $\Phi$-variation norm of $X$. For an Hermite process $X$ of order $m\in \N$ and of Hurst index $H\in (1/2,1)$, we show that $X$ is of bounded $\Phi$-variation where $\Phi(x)=x^{1/H}(\log(\log 1/x))^{-m/(2H)}$, and that this $\Phi$ is optimal. This shows that in terms of $\Phi$-variation, the Rosenblatt process (corresponding to $m=2$) has more rough sample paths than the fractional Brownian motion (corresponding to $m=1$).