Hausdorff–Besicovitch dimension of graphs and $p$-variation of some Lévy processes

Abstract

The connection between Hausdorff–Besicovitch dimension of graphs of trajectories and various Blumenthal–Getoor indices is well known for $α$-stable Lévy processes as well as for some stationary Gaussian processes possessing Orey index. We show that the same relationship holds for several classes of Lévy processes that are popular in financial mathematics models – in particular, the Carr–Geman–Madan–Yor, normal inverse Gaussian, generalized hyperbolic, generalized $z$ and Meixner processes.