On the Ф‐variation of stochastic processes with exponential moments

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