Behaviour of linear multifractional stable motion: membership of a critical Hölder space

Abstract

The study of path behaviour of stochastic processes is a classical topic in probability theory and related areas. In this frame, a natural question one can address is: whether or not sample paths belong to a critical Hölder space? The answer to this question is negative in the case of Brownian motion and many other stochastic processes: it is well-known that despite the fact that Brownian paths satisfy, on each compact interval I, a Hölder condition of any order strictly less than 1/2, they fail to belong to the critical Hölder space C 1/2 (I). In this article, we show that a different phenomenon happens in the case of linear multifractional stable motion (LMSM): for any given compact interval one can find a critical Hölder space to which sample paths belong. Among other things, this result improves an upper estimate, recently derived in (Biermé, H. and Lacaux, C., 2013), on behaviour of LMSM, by showing that the logarithmic factor in it is not needed. (2010): 60G22, 60G52, 60G17. Mathematics Subject Classification

Multifractional Hermite processes: definition and first properties

“…Plugging this into (11) and using again the isometry property (3) for Wiener-Itô integrals, we get Corollary 2.3. Given d ∈ N * and K a compact set of ( 1 2 , 1), let I be a compact interval of R + .…”

“…Also, different generalizations has been given such as in [13,14,3], where a larger class of Hurst functions are considered, in order that the Hölder exponent of the process is, almost surely, of the most general form given in [1,26], or in [15,12,4] where the Hurst function is also random. Finally, various extensions have been given, using larger classes of processes closely related to the fractional Brownian motion like, for instance, the linear multifractional stable motion [10,11] or the Surgailis multifractional process [57,7]. We also refer to the book [5] for a very clear view on the known facts about multifractional Brownian motion and related fields.…”