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Incremental moments and Hölder exponents of multifractional multistable processes

Published online by Cambridge University Press:  08 February 2013

Ronan Le Guével  and
Jacques Lévy Véhel
Ronan Le Guével
Affiliation:
UniversitéParis VI, Laboratoire de Probabilités et Modèles Aléatoires 4 place Jussieu, 75252 Paris Cedex 05, France. ronan.leguevel@upmc.fr
Jacques Lévy Véhel
Affiliation:
Regularity team, INRIA Saclay, Parc Orsay Université 4 rue Jacques Monod, Bat P, 91893 Orsay Cedex, France; jacques.levy-vehel@inria.fr
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Abstract

Multistable processes, that is, processes which are, at each “time”, tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise Hölder exponent is, as expected, related to the local stability index. We compute the precise value of the almost sure Hölder exponent in the case of the multistable Lévy motion, which turns out to reveal an interesting phenomenon.

Keywords

Localisable processesmultistable processesmultifractional processespointwise Hölder regularity
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 135 - 178
Copyright
© EDP Sciences, SMAI, 2013

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