arXiv subjects

Laurent Loosveldt

Publications and source records attributed to Laurent Loosveldt.

13 recordsLinked to original sources

arXiv2026

Malliavin calculus and densities for chaos-driven stochastic differential equations

We study stochastic differential equations driven by finite-order chaos processes on abstract Wiener spaces, with pathwise Riemann-Stieltjes integration. The driving noise is an $\mathbb{R}^m$-valued chaotic process given by multiple Wiener-Itô integrals of fixed order, allowing for non-Gaussian dynamics. Under mild smoothness assumptions on the coefficients and Hölder-type regularity of the noise, we establish existence and uniqueness of solutions. We then prove Malliavin differentiability and absolute continuity of the law of the solution. Since the usual Gaussian isonormal framework is unavailable, we rely on the Kusuoka-Stroock approach to Malliavin calculus and develop a Taylor expansion for multiple integrals under Cameron-Martin shifts. Under suitable ellipticity, independence, and non-degeneracy conditions, the Bouleau-Hirsch criterion yields density results. Applications to multidimensional Hermite-driven equations are provided.

math.PR↗

arXiv2025

Malliavin smoothness of the Rosenblatt process

We investigate the smoothness of the densities of the finite-dimensional distributions of the Rosenblatt process. Within the Malliavin calculus framework, we prove that Rosenblatt random vectors are nondegenerate in the Malliavin sense. As a consequence, their densities belong to the Schwartz space of rapidly decreasing smooth functions. The proof relies on establishing the existence of all negative moments of the determinant of the Malliavin matrix, exploiting the specific structure of random variables in the second Wiener chaos. In addition, we derive exponential-type upper bounds for the partial derivatives of the densities of the finite-dimensional distributions of the Rosenblatt process.

math.PR↗

arXiv2025

Absolute continuity of finite-dimensional distributions of Hermite processes via Malliavin calculus

We investigate the existence of densities for finite-dimensional distributions of Hermite processes of order $q \ge 1$ and self-similarity parameter $H\in(\frac12,1)$. Whereas the Gaussian case $q=1$ (fractional Brownian motion) is well understood, the non-Gaussian situation has not yet been settled. In this work, we extend the classical three-step approach used in the Gaussian case: factorization of the determinant into conditional terms, strong local nondeterminism, and non-degeneracy. We transport this strategy to the Hermite setting using Malliavin calculus. Specifically, we establish a determinant identity for the Malliavin matrix, prove strong local nondeterminism at the level of Malliavin derivatives, and apply the Bouleau-Hirsch criterion. Consequently, for any distinct times $t_1,\dots,t_n$, the vector $(Z^{H,q}_{t_1},\dots,Z^{H,q}_{t_n})$ of a Hermite process admits a density with respect to the Lebesgue measure. Beyond the result itself, the main contribution is the methodology, which could extend to other non-Gaussian models.

math.PR↗

arXiv2025

Weighted tensorized fractional Brownian textures

This paper presents a new model of textures, obtained as realizations of a new class of fractional Brownian fields. These fields, called weighted tensorized fractional Brownian fields, are obtained by a relaxation of the tensor-product structure that appears in the definition of fractional Brownian sheets. Statistical properties such as self-similarity, stationarity of rectangular increments and regularity properties are obtained. An operator scaling extension is defined and we provide simulations of the fields using their spectral representation.

math.PR↗

arXiv2024

Regularity of Weighted Tensorized Fractional Brownian Fields and associated function spaces

We investigate a new class of self-similar fractional Brownian fields, called Weighted Tensorized Fractional Brownian Fields (WTFBS). These fields, introduced in the companion paper \cite{ELLV}, generalize the well-known fractional Brownian sheet (FBs) by relaxing its tensor-product structure, resulting in new self-similar Gaussian fields with stationary rectangular increments that differ from the FBs. We analyze the local regularity properties of these fields and introduce a new concept of regularity through the definition of Weighted Tensorized Besov Spaces. These spaces combine aspects of mixed dominating smoothness spaces and hyperbolic Besov spaces, which are similar in structure to classical Besov spaces. We provide a detailed characterization of these spaces using Littlewood-Paley theory and hyperbolic wavelet analysis.

math.PR↗

arXiv2024

Modified wavelet variation for the Hermite processes

We define an asymptotically normal wavelet-based strongly consistent estimator for the Hurst parameter of any Hermite processes. This estimator is obtained by considering a modified wavelet variation in which coefficients are wisely chosen to be, up to negligeable remainders, independent. We use Stein-Malliavin calculus to prove that this wavelet variation satisfies a multidimensional Central Limit Theorem, with an explicit bound for the Wasserstein distance.

math.ST↗

arXiv2023

Wavelet-Type Expansion of Generalized Hermite Processes with rate of convergence

Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and many other related stochastic processes and fields have started to be introduced since more than two decades. Such representations provide natural frameworks for approximating almost surely and uniformly rough sample paths at different scales and for study of various aspects of their complex erratic behavior. Hermite process of an arbitrary integer order $d$, which extends FBM, is a paradigmatic example of a stochastic process belonging to the $d$th Wiener chaos. It was introduced very long time ago, yet many of its properties are still unknown when $d\ge 3$. In a paper published in 2004, Pipiras raised the problem to know whether wavelet-type random series representations with a well-localized smooth scaling function, reminiscent to those for FBM due to Meyer, Sellan and Taqqu, can be obtained for a Hermite process of any order $d$. He solved it in this same paper in the particular case $d=2$ in which the Hermite process is called the Rosenblatt process. Yet, the problem remains unsolved in the general case $d\ge 3$. The main goal of our article is to solve it, not only for usual Hermite processes but also for generalizations of them. Another important goal of our article is to derive almost sure uniform estimates of the errors related with approximations of such processes by scaling functions parts of their wavelet-type random series representations.

math.PR↗

arXiv2023

Multifractional Hermite processes: definition and first properties

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener-Itô integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener chaos, using facts from Malliavin calculus.

math.PR↗

arXiv2023

On the pointwise regularity of the Multifractional Brownian Motion and some extensions

We study the pointwise regularity of the Multifractional Brownian Motion and in particular, we get the existence of slow points. It shows that a non self-similar process can still enjoy this property. We also consider various extensions of our results in the aim of requesting a weaker regularity assumption for the Hurst function without altering the regularity of the process.

math.PR↗

arXiv2022

Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

We prove that we can identify three types of pointwise behaviour in the regularity of the (generalized) Rosenblatt process. This extends to a non Gaussian setting previous results known for the (fractional) Brownian motion. On this purpose, fine bounds on the increments of the Rosenblatt process are needed. Our analysis is essentially based on various wavelet methods.

math.PR↗

arXiv2022

Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions

We study the Hölderian regularity of Gaussian wavelets series and show that they display, almost surely, three types of points: slow, ordinary and rapid. In particular, this fact holds for the Fractional Brownian Motion. We also show that this property is satisfied for a multifractal extension of Gaussian wavelet series. Finally, we remark that the existence of slow points is specific to these functions.

math.PR↗

arXiv2020

Continuously differentiable functions on compact sets

We consider the space of real-valued continuously differentiable functions on a compact subset of a euclidean space. We characterize the completeness of this space and prove that the space of restrictions of continuously differentiable functions on the ambient space is always dense. The space is then compared with other spaces of differentiable functions on compact sets.

math.CA↗

arXiv2019

Generalized spaces of pointwise regularity: To a general framework for the WLM

In this work we generalize the spaces T^{p}_{u} introduced by Calderón and Zygmund using a pointwise version of conditions defining the generalized Besov spaces and give conditions binding the functions belonging to these spaces and the wavelet coefficients of such functions. Next, we propose a multifractal formalism based on the new spaces wich generalize the so-called wavelet leaders method and show that it is satisfied on a prevalent set.

math.FA↗