On the Ф‐variation of stochastic processes with exponential moments

Abstract

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)=x1/H(log(log1/x))−m/(2H) , 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 ).