Rosenblatt Process Research Articles

Approximate controllability of nonlinear Hilfer fractional stochastic differential system with Rosenblatt process and Poisson jumps

Abstract This manuscript is concerned with the approximate controllability problem of Hilfer fractional stochastic differential system (HFSDS) with Rosenblatt process and Poisson jumps. We derive the main results in stochastic settings by employing analytic resolvent operators, fractional calculus and fixed point theory. Further, we express the theoretical result with an example.

Wavelet-Type Expansion of the Generalized Rosenblatt Process and Its Rate of Convergence

Pipiras introduced in the early 2000s an almost surely and uniformly convergent (on compact intervals) wavelet-type expansion of the classical Rosenblatt process. Yet, the issue of estimating, almost surely, its uniform rate of convergence remained an open question. The main goal of our present article is to provide an answer to it in the more general framework of the generalized Rosenblatt process, under the assumption that the underlying wavelet basis belongs to the class due to Meyer. The main ingredient of our strategy consists in expressing in a non-classical (new) way the approximation errors related with the approximation spaces of a multiresolution analysis of $$L^2({{\mathbb {R}}}^2)$$. Such a non-classical expression may also be of interest in its own right.

Limit theorems in the context of multivariate long-range dependence

This article considers multivariate linear processes whose components are either short- or long-range dependent. The functional central limit theorems for the sample mean and the sample autocovariances for these processes are investigated, paying special attention to the mixed cases of short- and long-range dependent series. The resulting limit processes can involve multivariate Brownian motion marginals, operator fractional Brownian motions and matrix-valued versions of the so-called Rosenblatt process.

Non symmetric Rosenblatt process over a compact

In this short note, we give the representation of the non symmetric Rosenblatt process as a Wiener–Itô multiple integral with respect to the Brownian motion on a finite interval. Based on this representation, we obtain a least square-type estimator for an unknown parameter of the drift coefficient of a simple model driven by the non symmetric Rosenblatt process.

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.

A stochastic calculus for Rosenblatt processes

A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for Itô processes. These processes for this stochastic calculus arise naturally from a stochastic chain rule for functionals of Rosenblatt processes; and some Itô-type expressions are given here. Furthermore, there is some analysis of these results for their applications to problems using Rosenblatt noise.

Exponential behavior of neutral impulsive stochastic integro-differential equations driven by Poisson jumps and Rosenblatt process

AbstractIn this article, we are concerned with the neutral impulsive stochastic integro-differential equations driven by Poisson jumps and Rosenblatt process. By using resolvent operator and some analysis techniques, we ensure existence and uniqueness of solutions. Further, we investigate exponential stability of mild solutions. We have also given an example to illustrate our theoretical results.

Ergodic Linear-Quadratic Control for a Two Dimensional Stochastic System Driven by a Continuous Non-Gaussian Noise

Abstract In this paper an infinite time horizon (ergodic) quadratic cost control problem for a linear two dimensional stochastic system with a two dimensional Rosenblatt noise process is solved by providing an explicit expression to determine the optimal feedback. The system has some symmetry properties that allow for an explicit determination of an optimal control. The controls are the family of constant linear feedbacks which is known to be the natural family of controls for a Brownian motion noise to determine optimality. This family of constant linear feedback controls allows for practical implementation of the optimal control. Rosenblatt processes are continuous, non-Gaussian processes that have a long range dependence and a useful stochastic calculus and they are generated by double Wiener-Ito integrals with singular kernels. The long range dependence property of the Rosenblatt processes is a natural generalization from an important subfamily of (Gaussian) fractional Brownian motions. Long range dependent processes have been identified empirically in a significant variety of physical phenomena. An expression is obtained to determine explicitly the optimal ergodic control. The ergodic control result in this paper seems to be the first explicit ergodic control result for a multidimensional control system with a continuous, non-Gaussian noise. Furthermore it seems to be the first solution for a multi-dimensional game problem with a Rosenblatt noise.

Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields

Let Zt q,H t∈[0,1]d denote a d-parameter Hermite random field of order q ≥ 1 and self-similarity parameter H = H₁, . . . ,Hd ∈ ½, 1d. This process is H-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion q = 1, d = 1, fractional Brownian sheet q = 1, d ≥ 2, the Rosenblatt process q = 2, d = 1 as well as the Rosenblatt sheet q = 2, d ≥ 2. For any q ≥ 2, d ≥ 1 and H ∈ ½, 1d we show in this paper that a proper renormalization of the quadratic variation of Zq,H converges in L2Ω to a standard d-parameter Rosenblatt random variable with self-similarity index H' = 1 + 2H − 2/q.

Stability result of higher-order fractional neutral stochastic differential system with infinite delay driven by Poisson jumps and Rosenblatt process

In this manuscript, we establish the stability results of higher-order fractional neutral stochastic differential system (FNSDs) with infinite delay driven by Poisson jumps and Rosenblatt process via the Nussbaum fixed point theorem. The results are obtained by a new integral inequality. The validation of the efficiency of the proposed method is done through a numerical example.

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.

Pth Moment stability of fractional stochastic differential inclusions via resolvent operators driven by the Rosenblatt process and poisson jumps with impulses

ABSTRACT This study is concerned with the existence of mild solution for the analysis of the moment stability of fractional stochastic differential inclusions driven by the Rosenblatt process and Poisson jumps with impulses in a Hilbert space. The results are established by using the fixed point theorem, multi-valued functions, and fractional calculus. Finally, numerical simulation is implemented to verify the attained theoretical results.

Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process

The existence and uniqueness of mild solution of an impulsive stochastic system driven by a Rosenblatt process is analyzed in this work by using the Banach fixed point theorem and the theory of resolvent operator developed by R. Grimmer in [ 12 ]. Furthermore, the exponential stability in mean square for the mild solution to neutral stochastic integro–differential equations with Rosenblatt process is obtained by establishing an integral inequality. Finally, an example is exhibited to illustrate the abstract theory.

Neutral stochastic functional differential evolution equations driven by Rosenblatt process with varying-time delays

Hermite processes are self-similar processes with stationary increments, the Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by Rosenblatt process with index H ∈ ( 1/2 , 1) which is a special case of a self-similar process with long-range dependence. More precisely, we prove the existence and uniqueness of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.

Limit behavior of the Rosenblatt Ornstein–Uhlenbeck process with respect to the Hurst index

We study the convergence in distribution, as H → 1 2 and as H → 1, of the integral R f (u)dZ H (u), where Z H is a Rosenblatt process with self-similarity index H ∈ 1 2 , 1 and f is a suitable deterministic function. We focus our analysis on the case of the Rosenblatt Ornstein-Uhlenbeck process, which is the solution of the Langevin equation driven by the Rosenblatt process.

On impulsive Hilfer fractional stochastic differential system driven by Rosenblatt process

This manuscript investigates a class of impulsive Hilfer fractional stochastic differential system driven by Rosenblatt process with index 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 (fBm) and of order 2 is the Rosenblatt process. We establish new criteria to guarantee the existence of mild solutions in the stochastic settings by utilizing the Darbo-Sadovskii fixed point theorem, fractional calculus, Hausdorff measure of noncompactness and evolution operators. The derived result in this article is new in the sense that it generalizes many of the existing results in the literature, more precisely for Rosenblatt process and impulsive case of Hilfer derivative in the stochastic settings. Finally, stochastic partial differential equations are provided to validate the applicability of the derived theoretical results.

Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results.

Parameter estimation for the Rosenblatt Ornstein–Uhlenbeck process with periodic mean

We study the least squares estimator for the drift parameter of the Langevin stochastic equation driven by the Rosenblatt process. Using the techniques of the Malliavin calculus and the stochastic integration with respect to the Rosenblatt process, we analyze the consistency and the asymptotic distribution of this estimator. We also introduce alternative estimators, which can be simulated, and we study their asymptotic properties.

Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process

In this paper, we investigate the controllability and exponential stability results for a class of nonlinear neutral stochastic functional differential control systems in the presence of infinite delay driven by Rosenblatt process in real separable Hilbert spaces. Firstly, by using stochastic analysis approach, fractional calculus theory and a fixed point technique, we prove the controllability result for the mild solutions of the nonlinear control systems with the condition that the associated linear system is controllable. Secondly, we discuss the exponential stability of mild solutions of the nonlinear neutral stochastic differential systems driven by Rosenblatt process. Finally, an example is presented to illustrate the obtained theory.

Particle picture representation of the non-symmetric Rosenblatt process and Hermite processes of any order

We provide a particle picture representation for the non-symmetric Rosenblatt process and for Hermite processes of any order, extending the result of Bojdecki, Gorostiza and Talarczyk in [4]. We show that these processes can be obtained as limits of certain functionals of a system of particles evolving according to symmetric stable Lévy motions. In the case of [Formula: see text]-Hermite processes the corresponding functional involves [Formula: see text]-intersection local time of symmetric stable Lévy processes.