Hermite Process Research Articles

Wavelet-Type Expansion of Generalized Hermite Processes with Rate of Convergence

Wavelet-Type Expansion of Generalized Hermite Processes with Rate of Convergence

Asymptotic normality for a modified quadratic variation of the Hermite process

Asymptotic normality for a modified quadratic variation of the Hermite process

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.

The overdamped generalized Langevin equation with Hermite noise

We consider the generalized Langevin equation driven by a fractional non-Gaussian noise, the so-called Hermite process. We prove the existence and the uniqueness of the solution for this equation as well as various properties. An useful link between the solution to the Langevin equation with Hermite noise and a class of self-similar stochastic processes with stationary increments is given in our work. Based on this relationship, we deduce the regularity of the sample paths of the solution and the behavior of its p-variation. Some applications concerning the statistical estimation of the noise intensity parameter or the Euler method for the generalized Langevin equation are also discussed.

Wiener integrals with respect to the generalized Hermite process (gHp). Applications: SDEs with gHp noise

Abstract In this paper, we introduce Wiener integrals with respect to the generalized Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an application, we investigate the corresponding stochastic differential equations with the generalized Hermite process as a driving noise, we prove the existence and the uniqueness of the solution, and we give a generalization of the Hermite Ornstein–Uhlenbeck process and the Hermite-driving Vasicek process.

Non-Gaussian Traffic Modeling for Multicore Architecture Using Wavelet Based Rosenblatt Process

Networks-on-Chip (NoCs) provides a packet-based communication model for the System-on-Chip (SoC) architecture to achieve objectives of lower power, higher performance and optimized area. To evaluate NoC performance and explore architectural design options, synthetic traffic models provide better solution in terms of simulation flexibility and faster simulation. In this synthetic traffic, models are generated for the simulation to optimize the computational resources requirements which will be placed inside NoC’s architecture before the early design process. In this paper, Wavelet-based Rosenblatt synthetic traffic [RST] is designed and examined which is a non-Gaussian process based on the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>nd</i></sup> -order Hermite Process. The proposed traffic is applied to SPLASH-2 benchmarks and compared with Gaussian and Non-Gaussian traffic on the parameters of latency, power and buffer-loss. We applied the synthetic traffic on multicore architecture to capture parameters which would be helpful for network designers to choose optimized network resources during the early design architecture. We demonstrate our traffic model: first, how it can be used to describe and acquire non-gaussian burstier traffic on NoC; second, how it can be used to produce synthetic traffic traces that can drive exploration of the early design space for NoCs. Our traffic model helps the chip network designer to choose optimized queue storage inside the router in early design process.

Arbitrage in the Hermite Binomial Market

Much attention has been paid to the arbitrage opportunities in the Black–Scholes model when it is driven by fractional Brownian motions. It is natural to ask whether there exists arbitrage or not when we focus on other fractional processes, such as the Hermite process. We set forth an approximation of the Hermite Black–Scholes model by random walks in the Skorokhod topology, and apply the Donsker type approximation to the Hermite process as the Hurst index is greater than 12. We find that the binary model approximation of the Black–Scholes model driven by Hermite processes also admits arbitrage opportunities. Several numerical examples of the Hermite binomial model are presented as demonstration. Moreover, we provide an option pricing model when the geometric Hermite processes drives the price fluctuations of the underlying asset.

Besov-Orlicz Path Regularity of Non-Gaussian Processes

In the article, Besov-Orlicz regularity of sample paths of stochastic processes that are represented by multiple integrals of order \(n\in \mathbb {N}\) is treated. We assume that the considered processes belong to the Hölder space $$ C^{\alpha}([0,T];L^{2}({\Omega}))\quad\text{with}\quad \alpha\in (0,1), $$ and we give sufficient conditions for them to have paths in the exponential Besov-Orlicz space $$ B_{{\varPhi}_{2/n},\infty}^{\alpha}(0,T)\qquad \text{with}\qquad {\varPhi}_{2/n}(x)=\mathrm{e}^{x^{2/n}}-1. $$ These results provide an extension of what is known for scalar Gaussian stochastic processes to stochastic processes in an arbitrary finite Wiener chaos. As an application, the Besov-Orlicz path regularity of fractionally filtered Hermite processes is studied. But while the main focus is on the non-Gaussian case, some new path properties are obtained even for fractional Brownian motions.

Generalized Wiener–Hermite integrals and rough non-Gaussian Ornstein–Uhlenbeck process

We discuss several properties of the generalized Hermite process, which is a non-Gaussian self-similar processes with self-similarity index belonging to the whole interval . We also define Wiener integral with respect to this process and as an application, we define and study its associated Ornstein–Uhlenbeck process.

Stochastic integration with respect to fractional processes in Banach spaces

In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is considered includes, for example, fractional Brownian motions of any Hurst parameter or, more generally, fractionally filtered generalized Hermite processes. The class of Banach spaces that is considered includes a large variety of the most commonly used function spaces such as the Lebesgue spaces, Sobolev spaces, or, more generally, the Besov and Lizorkin-Triebel spaces. In the article, a characterization of the domains of the Wiener integrals on both bounded and unbounded intervals is given for both scalar and cylindrical fractional processes. In general, the integrand takes values in the space of γ-radonifying operators from a certain homogeneous Sobolev-Slobodeckii space into the considered Banach space. Moreover, an equivalent characterization in terms of a pointwise kernel of the integrand is also given if the considered Banach space is isomorphic with a subspace of a cartesian product of mixed Lebesgue spaces. The results are subsequently applied to stochastic convolution for which both necessary and sufficient conditions for measurability and sufficient conditions for continuity are found. As an application, space-time continuity of the solution to a parabolic equation of order 2m with distributed noise of low time regularity is shown as well as measurability of the solution to the heat equation with Neumann boundary noise of higher regularity.

Null controllability of nonlocal Sobolev-Type Hilfer fractional stochastic differential system driven by fractional Brownian motion and Poisson jumps

In this manuscript, we establish a class of nonlocal Sobolev-type Hilfer fractional stochastic differential equations driven by fractional Brownian motion, which is a special case of a self-similar process, Hermite processes with stationary increments with long-range dependence. The Hermite process of order 1 is fractional Brownian motion and of order 2 is the Rosenblatt process. By using fractional calculus and fixed point approach, sufficient conditions of exact null controllability for such fractional stochastic systems are established. The derived result in this manuscript is new in the sense that it generalizes many of the existing results in the literature, more precisely for fractional Brownian motion and Poisson jumps case of Sobolev-type Hilfer fractional stochastic settings. Finally, stochastic partial differential equations are provided to validate the applicability of the derived theoretical results.

Limit theorems for integral functionals of Hermite-driven processes

Consider a moving average process X of the form X(t)=∫−∞tφ(t−u)dZu, t≥0, where Z is a (non Gaussian) Hermite process of order q≥2 and φ:R+→R is sufficiently integrable. This paper investigates the fluctuations, as T→∞, of integral functionals of the form t↦∫0TtP(X(s))ds, in the case where P is any given polynomial function. It extends a study initiated in (Stoch. Dyn. 18 (2018) 1850028, 18), where only the quadratic case P(x)=x2 and the convergence in the sense of finite-dimensional distributions were considered.

Functional limit theorems for power series with rapid decay of moving averages of Hermite processes

We aim to obtain a homogenization theorem for a passive tracer interacting with a fractional, possibly non-Gaussian, noise. To do so, we analyze limit theorems for normalized functionals of Hermite–Volterra processes and extend existing results to cover power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short- and long-range dependent components. Applying this theorem, we obtain a homogenization result for a slow/fast system driven by such Hermite noises.

The Least Squares Estimator for an Ornstein-Uhlenbeck Process Driven by a Hermite Process with a Periodic Mean

We consider the least square estimator for the parameters of Ornstein-Uhlenbeck processes $$d{Y_s} = \left( {\sum\limits_{j = 1}^k {{\mu _j}{\phi _j}\left( s \right) - \beta {Y_s}} } \right){\rm{d}}s + {\rm{dZ}}_s^{q,H},$$ driven by the Hermite process Z with order q ≥ 1 and a Hurst index H ∈ (½, 1), where the periodic functions φj(s),j = 1,⌦, k are bounded, and the real numbers μj, j = 1, …, k together with β > 0 are unknown parameters. We establish the consistency of a least squares estimation and obtain the asymptotic behavior for the estimator. We also introduce alternative estimators, which can be looked upon as an application of the least squares estimator. In terms of the fractional Ornstein-Uhlenbeck processes with periodic mean, our work can be regarded as its non-Gaussian extension.

Representations of hermite processes using local time of intersecting stationary stable regenerative sets

AbstractHermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.

On one interpolating rational process of Fejer – Hermite

In this paper, a new approach to the definition of the interpolating rational process of Fejer – Hermite with first-type Chebyshev – Markov nodes on a segment is studied and some of its approximating properties are described. In the introduction a brief analysis of the results on the topic of the research is carried out. Herein, the methods of the construction of interpolating processes, in particular, Fejer – Hermite processes, in the polynomial and rational approximation are analysed. A new method to determine the interpolating rational Fejer – Hermite process is proposed. One of the main results of this paper is the proof of the uniform convergence of this process for an arbitrary function, which is continuous on the segment, under some restrictions for the poles of approximating functions. This result is preceded by some auxiliary statements describing the properties of special rational functions. The classic methods of mathematical analysis, approximation theory, and theory of functions of a complex variable are used to prove the results of the work. Moreover, we present the numerical analysis of the effectiveness of the application of the constructed interpolating Fejer – Hermite process for the approximation of a continuous function with singularities. The choice of parameters, on which the nodes of interpolation depend, is made in several standard ways. The obtained results can be applied to further study the approximating properties of interpolating processes.

Parameter identification for the Hermite Ornstein–Uhlenbeck process

By using the analysis on Wiener chaos, we study the behavior of the quadratic variations of the Hermite Ornstein–Uhlenbeck process, which is the solution to the Langevin equation driven by a Hermite process. We apply our results to the identification of the Hurst parameter of the Hermite Ornstein–Uhlenbeck process.

On Estimating the Hurst Parameter from Least-Squares Residuals. Case Study: Correlated Terrestrial Laser Scanner Range Noise

Many signals appear fractal and have self-similarity over a large range of their power spectral densities. They can be described by so-called Hermite processes, among which the first order one is called fractional Brownian motion (fBm), and has a wide range of applications. The fractional Gaussian noise (fGn) series is the successive differences between elements of a fBm series; they are stationary and completely characterized by two parameters: the variance, and the Hurst coefficient (H). From physical considerations, the fGn could be used to model the noise of observations coming from sensors working with, e.g., phase differences: due to the high recording rate, temporal correlations are expected to have long range dependency (LRD), decaying hyperbolically rather than exponentially. For the rigorous testing of deformations detected with terrestrial laser scanners (TLS), the correct determination of the correlation structure of the observations is mandatory. In this study, we show that the residuals from surface approximations with regression B-splines from simulated TLS data allow the estimation of the Hurst parameter of a known correlated input noise. We derive a simple procedure to filter the residuals in the presence of additional white noise or low frequencies. Our methodology can be applied to any kind of residuals, where the presence of additional noise and/or biases due to short samples or inaccurate functional modeling make the estimation of the Hurst coefficient with usual methods, such as maximum likelihood estimators, imprecise. We demonstrate the feasibility of our proposal with real observations from a white plate scanned by a TLS.

Lower bound for local oscillations of Hermite processes

The most known example of a class of non-Gaussian stochastic processes which belongs to the homogeneous Wiener chaos of an arbitrary order N>1 is probably Hermite processes of rank N. They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable functions has not yet been solved even in the most simple case of the Rosenblatt process. The goal of our article is to derive a quasi-optimal lower bound for the asymptotic behavior of local oscillations of paths of Hermite processes of any rank N, which, among other things, shows that these paths are nowhere differentiable functions.

Existence, uniqueness and stability of impulsive stochastic neutral functional differential equations driven by Rosenblatt process with varying-time delays

Abstract Hermite processes are self-similar processes with stationary increments; the Hermite process of order 1 is fractional Brownian motion (fBm) and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of impulsive neutral stochastic functional differential equations with variable delays driven by Rosenblatt process with index {H\in(\frac{1}{2},1)} , which is a special case of a self-similar process with long-range dependence. More precisely, we prove an existence and uniqueness result, and we establish some conditions, ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point theory. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.