Random Wavelet Series Based on a Tree-Indexed Markov Chain

Durand, Arnaud

doi:10.1007/s00220-008-0504-7

Cited by 17 publications

(24 citation statements)

References 38 publications

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“…When the function f is globally Hölder continuous, it has been proved in [25,31] that the exponent h f (t) can always be obtained through the asymptotic behavior of the wavelet coefficients of f located in a neighborhood of t, when the wavelet is smooth enough. Then, wavelet expansions have been used successfully to characterize the iso-Hölder sets of wide classes of functions [26,27,11,29,30,3,9,17], sometimes directly constructed as wavelet series (expansions in Schauder's basis have also been used [32]). …”

Section: Introductionmentioning

confidence: 99%

Multifractal Analysis of Complex Random Cascades

Barral

Jin

2010

Commun. Math. Phys.

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Abstract. We achieve the multifractal analysis of a class of complex valued statistically self-similar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new phenomena in multifractal analysis of continuous functions. In particular, we find examples of statistically self-similar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval [0, ∞].

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Section: Introductionmentioning

confidence: 99%

Multifractal Analysis of Complex Random Cascades

Barral

Jin

2010

Commun. Math. Phys.

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“…the Hurst exponent H in the case of f.B.m.). Nevertheless, note that Barral et al [11] and Durand [17] have provided examples of respectively Markov jump processes and wavelet random series with a non-homogeneous and random spectrum of singularities.As outlined in Equations (1.1) and (1.2), multifractal analysis usually focuses on the structure of pointwise regularity. Unfortunately, as presented by Meyer [37], the pointwise Hölder exponent suffers of a couple of drawbacks: it lacks of stability under the action of pseudo-differential operators and it is not always characterised by the wavelets coefficients.…”

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confidence: 99%

Fine regularity of Lévy processes and linear (multi)fractional stable motion

Balança

2014

Electron. J. Probab.

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International audienceIn this work, we investigate the fine regularity of Lévy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Lévy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional stable motion (extension of fractional Brownian motion to α-stable measures) in the case of continuous and unbounded sample paths as well. The regularity of its multi-fractional extension is also presented, indirectly providing an example of a stochastic process with a non-homogeneous and random multifractal spectrum

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“…where the right-hand side integral is bounded above by C(1 + |M u− | 2 ) ≤ 2C(1 + |M t | 2 ) due to linear growth condition (H1) and the continuity of M at t. Combining (18)- (19), one obtains that…”

Section: 31mentioning

confidence: 97%

Multifractality of jump diffusion processes

Yang¹

2018

Ann. Inst. H. Poincaré Probab. Statist.

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We study the local regularity and multifractal nature of the sample paths of jump diffusion processes, which are solutions to a class of stochastic differential equations with jumps. This article extends the recent work of Barral et al. who constructed a pure jump monotone Markov process with random multifractal spectrum. The class of processes studied here is much larger and exhibits novel features on the extreme values of the spectrum. This class includes Bass' stable-like processes and non-degenerate stable-driven SDEs.Nousétudions la régularité locale et la nature multifractale des trajectoires de diffusionà sauts, qui sont solutions d' une classe d'équations stochastiquesà sauts. Cet article prolonge etétend substantiellement le travail récent de Barral et al. qui ont construit un processus de Markov de sauts purs avec un spectre multifractal aléatoire. La classe considérée est beaucoup plus large et présente de nouveaux phénomènes multifractals notamment sur les valeurs extrêmes du spectre. Cette classe comprend les processus de type stable au sens de Bass et des EDS non dégénérées guidées par un processus stable. ν γ (y, du) = γ(y)u −1−γ(y) 1 [0,1] du Date: September 24, 2018.

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Random Wavelet Series Based on a Tree-Indexed Markov Chain

Cited by 17 publications

References 38 publications

Multifractal Analysis of Complex Random Cascades

Multifractal Analysis of Complex Random Cascades

Fine regularity of Lévy processes and linear (multi)fractional stable motion

Multifractality of jump diffusion processes

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