Behaviour of linear multifractional stable motion: membership of a critical H\"older space

TL;DR

This paper investigates the path regularity of linear multifractional stable motion, demonstrating that unlike Brownian motion, LMSM sample paths can belong to a critical H"older space, revealing new regularity properties.

Contribution

It proves that LMSM sample paths belong to a critical H"older space, improving previous estimates by removing unnecessary logarithmic factors.

Findings

LMSM paths belong to a critical H"older space on any compact interval.

The result refines previous estimates by eliminating the logarithmic factor.

Contrasts LMSM behavior with Brownian motion regarding critical H"older spaces.

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 H\"older 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 H\"older condition of any order strictly less than $1/2$, they fail to belong to the critical H\"older space $\mathcal{C}^{1/2}(I)$. 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 H\"older space to which sample paths belong. Among other things, this result improves an upper estimate, recently derived in Bierm\'e, Lacaux (2013), on behaviour of LMSM, by showing that the logarithmic factor in it is not needed.