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

We obtain sharp sufficient conditions for exponentially integrable stochastic processes $X=\{X(t)\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}:t\in [0,1]\}$, to have sample paths with bounded $\mathrm{\Phi }$-variation. When $X$ is moreover Gaussian, we also provide a bound of the expectation of the associated $\mathrm{\Phi }$-variation norm of $X$. For an Hermite process $X$ of order $m\in \mathbb{N}$ and of Hurst index $H\in (1\mathrm{/}2,1)$, we show that $X$ is of bounded $\mathrm{\Phi }$-variation where $\Phi(x)=x^{1/H}(\log(\log 1/x))^{-m/(2H)}$,andthatthis$\Phi$isoptimal\mathrm{.}Thisshowsthatintermsof$\Phi$-variation,theRosenblattprocess(correspondingto$m=2$) has more rough sample paths than the fractional Brownian motion (corresponding to $m=1$).