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

Abstract

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.

The $$n$$n-term Approximation of Periodic Generalized Lévy Processes

“…The index β was introduced by R. Blumenthal and R.K. Getoor in [6] to characterize the behavior of a Lévy process around the origin. Since then, it has been used to characterize many local properties of such processes, including the spectrum of singularities in multifractal analysis [15,30] and local self-similarity [19]. More importantly for our use, β allows us to characterize Lévy process [39,9] and noises [20,18,2] in terms of Besov spaces 1 .…”

“…Generalized Lévy processes encompass Gaussian models and the family of compound-Poisson processes that are pure jump processes. They also include symmetric-alpha-stable (SαS) processes which maintain many of the desirable properties of Gaussian models; for example, they satisfy a generalized central-limit theorem [37,19].…”

Wavelet Compressibility of Compound Poisson Processes

“…Again, this empirical fact has a theoretical counterpart: it is linked with the fact that the statistics of finite variance compound Poisson processes are barely distinguishable from the ones of the Brownian motion at coarse scales. This has been formalized in [64] which states, when particularized to our case, that compound Poisson processes with finite variance converge to the Brownian motion…”

Entropic Compressibility of Lévy Processes

“…Getoor in 1961 to characterize the small-time behavior of Lévy processes [8]. Since then, it has been recognized as a key quantity to characterize the local Besov regularity [64,63,3] or the variations [59] of the sample paths, the Hausdorff dimension of the image set [42,20], moment estimates [48,18,25,46], the local self-similarity [26], or the local wavelet compressibility [28] of Lévy processes and their generalizations. We demonstrate in this paper that it also quantifies the asymptotic behavior of the entropy of Lévy processes.…”

“…In contrast, the characterization of all stable laws requires two additional parameters, including a skewness parameter, and is thus more involved (see [61,Definition 1.1.6]). It should be possible to generalize Theorem 4.3 to locally self-similar processes with possibly non-symmetric local limit, but this requires extending the currently used results from [26] to non-symmetric limits. Note however that the asymptotic entropy of any (possibly non-symmetric) stable random process was obtained in [32], and that Theorem 4.5 is valid for arbitrary Lévy processes.…”