Linear Multifractional Stable Motion: fine path properties

TL;DR

This paper thoroughly investigates the path properties of Linear Multifractional Stable Motion (LMSM), introducing wavelet series representations and resolving a conjecture about its continuity, while providing new bounds on its local regularity.

Contribution

It introduces wavelet series representations for LMSM and its derivatives, and proves the existence of continuous modifications, also establishing new bounds on local Hölder exponents.

Findings

Established the existence of a modification of LMSM with almost surely continuous paths.

Derived bounds for the local Hölder exponent of LMSM.

Provided a new optimal local modulus of continuity for LMSM.

Abstract

Linear Multifractional Stable Motion (LMSM), denoted by $\{Y(t):t\in\R\}$, has been introduced by Stoev and Taqqu in 2004-2005, by substituting to the constant Hurst parameter of a classical Linear Fractional Stable Motion (LFSM), a deterministic function $H(\cdot)$ depending on the time variable $t$; we always suppose $H(\cdot)$ to be continuous and with values in $(1/\al,1)$, also, in general we restrict its range to a compact interval. The main goal of our article is to make a comprehensive study of the local and asymptotic behavior of $\{Y(t):t\in\R\}$; to this end, one needs to derive fine path properties of $\{X(u,v) : (u,v)\in\R \times (1/\alpha,1)\}$, the field generating the latter process (i.e. one has $Y(t)=X(t,H(t))$ for all $t\in\R$). This leads us to introduce random wavelet series representations of $\{X(u,v) : (u,v)\in\R \times (1/\alpha,1)\}$ as well as of all its pathwise partial derivatives of any order with respect to $v$. Then our strategy consists in using wavelet methods. Among other things, we solve a conjecture of Stoev and Taqqu, concerning the existence for LMSM of a modification with almost surely continuous paths; moreover we provides some bounds for the local H\"older exponent of LMSM: namely, we obtain a quasi-optimal global modulus of continuity for it, and also an optimal local one. It is worth noticing that, even in the quite classical case of LFSM, the latter optimal local modulus of continuity provides a new result which was unknown so far.