Lévy Processes and Lévy White Noise as Tempered Distributions

Dalang, Robert C.; Humeau, Thomas

doi:10.48550/arxiv.1509.05274

Cited by 3 publications

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“…Indeed, polynomial weights are not increasing fast enough to compensate the erratic behavior of the white noise. This is consistent with the fact that a Lévy white noise with β 0 = 0 is not tempered [9].…”

Section: Discussion and Comparison With Known Resultssupporting

confidence: 88%

“…As we briefly saw in Section 2.1, the class of Lévy white noises over D ′ (R d ) is strictly larger than the one over S ′ (R d ). A Lévy white noise over D ′ (R d ) is also in S ′ (R d ) if and only if its Lévy exponent satisfies the Schwartz condition [9] or, equivalently, if and only if its Blumenthal-Getoor index β 0 is not 0. Until now, we have only considered Lévy white noises for which β 0 = 0.…”

Section: We Summarize the Situation By Wmentioning

confidence: 99%

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Multidimensional Lévy white noise in weighted Besov spaces

Fageot

Fallah

Unser

2017

Stochastic Processes and their Applications

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In this paper, we study the Besov regularity of a general d-dimensional Lévy white noise. More precisely, we describe new sample paths properties of a given noise in terms of weighted Besov spaces. In particular, we characterize the smoothness and integrability properties of the noise using the indices introduced by Blumenthal, Getoor, and Pruitt. Our techniques rely on wavelet methods and generalized moments estimates for Lévy noises.

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Section: Discussion and Comparison With Known Resultssupporting

confidence: 88%

Section: We Summarize the Situation By Wmentioning

confidence: 99%

Multidimensional Lévy white noise in weighted Besov spaces

Fageot

Fallah

Unser

2017

Stochastic Processes and their Applications

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show abstract

“…A Lévy white noise w is the weak derivative of the corresponding Lévy process X with identical Lévy exponent. This well-known fact has been rigorously shown in the sense of generalized random processes in [5]. A direct consequence is that τ p (X) = τ p (w) + 1, where w = X ′ .…”

Section: Introduction and Main Resultsmentioning

confidence: 74%

Wavelet Analysis of the Besov Regularity of Lévy White Noises

Aziznejad,

Fageot

2018

Preprint

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In this paper, we characterize the local smoothness and the asymptotic growth rate of Lévy white noises. We do so by identifying the weighted Besov spaces in which they are localized. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We deduce the critical local smoothness when the two indices coincide, which is true for symmetric-α-stable, compound Poisson and symmetric-gamma white noises. Second, we express the critical asymptotic growth rate in terms of the moments properties of the Lévy white noise. Previous analysis only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires to determine in which Besov spaces a given Lévy white noise is (almost surely) not. Our methods are based on the wavelet-domain characterization of Besov spaces and precise moment estimates for the wavelet coefficients of the noise.

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“…We have shown that the existence of a finite absolute moment is sufficient for w being tempered in [18,Theorem 3]. More recently, R. Dalang and T. Humeau have proved that this condition is also necessary [12,Theorem 3.13]. This provides a one-to-one correspondence between tempered Lévy noise and infinitely divisible random variable having a finite absolute moment, what justifies the following definition.…”

Section: Lévy White Noises and Infinite Divisibilitymentioning

confidence: 79%

Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises

Fageot

Unser

2018

J Theor Probab

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Correspondence to be sent to: julien.fageot@epfl.ch Consider a random process s solution of the stochastic differential equation Ls = w with L a homogeneous operator and w a multidimensional Lévy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on L and w such that a H s(·/a) converges in law to a non-trivial self-similar process for some H, when a → 0 (coarse-scale behavior) and a → ∞ (fine-scale behavior). The parameter H depends on the homogeneity order of the operator L and the Blumenthal-Getoor and Pruitt indices associated to the Lévy white noise w. Finally, we apply our general results to several notorious classes of random processes and random fields and illustrate our results on simulations of Lévy processes. 2 homogeneous of order 1. Our aim is to study the statistical behavior of the rescaling x → s(x/a) of a solution of (2) when a > 0 is varying. Our two main questions are:• What is the asymptotic behavior of s(·/a) when we zoom out the process (i.e., when a → 0)?• What is the local behavior when we zoom in (i.e., when a → ∞)?Our main contribution is to identify sufficient conditions such that the rescaling a H s(·/a) of a solution of (2) has a non-trivial self-similar asymptotic limit as a goes to 0 or ∞. When this limit exists, the parameter H is unique and depends essentially on the degree of homogeneity γ of L and on the Blumenthal-Getoor and Pruitt indices β ∞ and β 0 of w [8,44]. The indices β 0 and β ∞ are used in the literature to characterize the asymptotic and local behaviors of Lévy processes, and more generally Lévy-type processes [9]. We summarize the main results of our paper in Theorem 1.1. Precise definitions and rigorous statements are given later.Theorem 1.1. Let L be a γ-homogeneous operator and w a Lévy process with indices β ∞ and β 0 . Under some technical conditions, the solution s to the equation Ls = w has the following properties.

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Lévy Processes and Lévy White Noise as Tempered Distributions

Cited by 3 publications

References 0 publications

Multidimensional Lévy white noise in weighted Besov spaces

Multidimensional Lévy white noise in weighted Besov spaces

Wavelet Analysis of the Besov Regularity of Lévy White Noises

Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises

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