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Analyse par ondelettes du mouvement multifractionnaire stable linéaire

Abstract : Fractional Brownian Motion (FBM) is an important tool in modeling used in several areas (biology, economics, finance, geology, hydrology, telecommunications, and so on); however, this model does not always give a sufficiently accurate description of reality, two important ones among its limitations, are the following: on one hand, FBM is a Gaussian process, and on the other hand, its local roughness (measured through a Hölder exponent) remains the same all along its path, since this roughness is everywhere equal to the Hurst parameter H which is a constant. In order to overcome the latter two limitations, S. Stoev and M.S. Taqqu (2004 and 2005) introduced Linear Multifractional Stable Motion (LMSM); this strictly α-stable (StαS) stochastic process, denoted by {Y(t)} , is obtained by replacing the Brownian measure by a StαS one and the Hurst parameter H by a function H(.) depending on t. One assumes the latter function to be continuous and with values in the open interval (1/α,1). Also, it is worth noticing that one has for all t, Y(t)=X(t,H(t)), where {X(u,v)} is the StαS stochastic field, such that for all fixed v, the process {X(u,v)} is a Linear Fractional Stable Motion. The goal of the thesis is to conduct a thorough study on LMSM by making use of wavelet methods; it mainly consists into three parts. (1) One determines, sharp global and local moduli of continuity for {Y(t)}; this mainly relies on a new representation of {X(u,v)}, as a random series which almost surely converges in some Hölder spaces. (2) One introduces, via the Haar basis, another random series representation of {X(u,v)}; the latter representation allows to derive an efficient simulation for LMSM as well as its high and low frequency parts. (3) One constructs wavelet estimators of the functional parameter H(.) of LMSM and of its stability parameter α.
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Submitted on : Monday, November 19, 2012 - 12:08:01 PM
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Julien Hamonier. Analyse par ondelettes du mouvement multifractionnaire stable linéaire. Probabilités [math.PR]. Université des Sciences et Technologie de Lille - Lille I, 2012. Français. ⟨tel-00753510⟩

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