Title :

Linear Multifractional Stable Motion: fine path properties

Author(s) :

Ayache, Antoine [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Hamonier, Julien [Auteur]
Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 [LAMAV]

English keyword(s) :

laws of the iterated logarithm
Linear Fractional and Multifractional Stable Motions
Wavelet series representations
Moduli of continuity
Holder regularity
laws of the iterated logarithm.

HAL domain(s) :

Mathématiques [math]/Probabilités [math.PR]

English abstract : [en]

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 ...
Show more >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 Hölder 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.Show less >

Language :

Anglais

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL