Stable Noises Research Articles

Existence and uniqueness of solutions for forward and backward nonlocal Fokker-Planck equations with time-dependent coefficients

Based on the Feynman-Kac formula, we develop a new approach to show the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations, which are associated with a class of time-dependent stochastic differential equations driven by symmetric (or asymmetric) non-Gaussian Lévy process. First, after deriving the forward time-dependent multidimensional nonlocal Fokker-Planck equation, the Feynman-Kac formula for the forward nonlocal Fokker-Planck equation is established by using Itô's formula and techniques for the backward nonlocal Fokker-Planck equation corresponding to the backward stochastic differential equation driven by Lévy process. Then, from the Feynman-Kac formula, we prove the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations by using techniques from stochastic analysis. Finally, the dynamic evolution of the probability density function corresponding to the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, comK, comS) is investigated over an extended period by Feynman-Kac formula and Monte Carlo simulations. In particular, we reveal the influence of symmetric and asymmetric stable noises as well as tempered stable noise on the probability density function corresponding to the stochastic MeKS network model.

Novel compound multistable stochastic resonance weak signal detection

Abstract The research on stochastic resonance (SR) which is used to extract weak signals from noisy backgrounds is of great theoretical significance and promising application. To address the shortcomings of the classical tristable SR model, this article proposes a novel compound multistable stochastic resonance (NCMSR) model by combining the Woods–Saxon (WS) and tristable models. The influence of the parameters of the NCMSR systems on the output response performance is studied under different α stable noises. Meanwhile, the adaptive synchronization optimization algorithm based on the proposed model is employed to achieve periodic and non-periodic signal identifications in α stable noise environments. The results show that the proposed system model outperforms the tristable system in terms of detection performance. Finally, the NCMSR model is applied to 2D image processing, which achieves great noise reduction and image recovery effects.

Generalized Hyperbolic Tangent Based Random Fourier Conjugate Gradient Filter for Nonlinear Active Noise Control

The filtered-x least mean square (FxLMS) algorithm has been proposed for an active noise control (ANC) system. However, due to the used mean square error (MSE) criterion, FxLMS suffers from performance degeneration for non-Gaussian noises, dramatically. To address this issue, a novel robust generalized hyperbolic tangent (GHT) criterion is first constructed in this paper. Then, the random Fourier features (RFF) method and the conjugate gradient (CG) method are used to address the nonlinearity existing in ANC and solve the quadratic optimization problem induced by the GHT criterion, respectively. Finally, a novel robust random Fourier conjugate gradient filtered-x generalized hyperbolic tangent (RFCGFxGHT) algorithm is proposed for ANC. The theoretical analyses regarding the convergence and computational complexity of RFCGFxGHT are also derived. Simulation experiments on nonlinear ANC systems corrupted by the synthetic logistic chaotic and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> -stable noises, as well as real-world functional magnetic resonance imaging (fMRI) and server room noises, are conducted to confirm the effectiveness, robustness, and desirable nonlinear learning ability of the proposed algorithm.

Drifted escape from the finite interval.

Properties of the noise-driven escape kinetics are mainly determined by the stochastic component of the system dynamics. Nevertheless, the escape dynamics is also sensitive to deterministic forces. Here, we are exploring properties of the overdamped drifted escape from finite intervals under the action of symmetric α-stable noises. We show that the properly rescaled mean first passage time follows the universal pattern as a function of the generalized Pécklet number, which can be used to efficiently discriminate between domains where drift or random force dominate. Stochastic driving of the α-stable type is capable of diminishing the significance of the drift in the regime when the drift prevails.

Performance Analysis of Fractional-Order Adaptive Filtering Algorithm and Its Improvement

Considering the <inline-formula><tex-math notation="LaTeX">$ \alpha$</tex-math></inline-formula>-stable input signals and noises, several robust fractional-order adaptive filtering algorithms have been proposed in recent years. However, the mean-square performance analysis of the FoAF algorithm has not been well studied in the literature. Toward that end, this letter provide a theoretical mean-square performance analysis of the FoAF algorithm, involving transient and steady-state behavior. Furthermore, an improved fractional-order adaptive filtering algorithm is derived by utilizing the M-estimate technique. Simulation experiments verify the results of the theoretical analysis and demonstrate the advantages of the proposed algorithm in heavy-tailed non-Gaussian environments.

Least squares estimation for discretely observed Ornstein–Uhlenbeck process driven by small stable noises

This article is concerned with the drift parameter estimation for Ornstein–Uhlenbeck process driven by small α-stable noises. The contrast function is given to obtain the least squares estimators and the error of estimation are obtained. The consistency, the rate of convergence and asymptotic distribution of estimators are derived when a small dispersion coefficient and simultaneously. Some numerical calculus and simulations are made to verify the effectiveness of the estimators.

Large deviation principle of occupation measures for non-linear monotone SPDEs

Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises.

Estimation for the Discretely Observed Cox–Ingersoll–Ross Model Driven by Small Symmetrical Stable Noises

This paper is concerned with the least squares estimation of drift parameters for the Cox–Ingersoll–Ross (CIR) model driven by small symmetrical α-stable noises from discrete observations. The contrast function is introduced to obtain the explicit formula of the estimators and the error of estimation is given. The consistency and the rate of convergence of the estimators are proved. The asymptotic distribution of the estimators is studied as well. Finally, some numerical calculus examples and simulations are given.

Wavelet analysis of the Besov regularity of Lévy white noise

We characterize the local smoothness and the asymptotic growth rate of the Levy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Levy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-$\alpha $-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the Levy white noise. Previous analyses only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires us to determine in which Besov spaces a given Levy 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 Levy white noise.

α-stable noise-induced coherence on a spatially extended Fitzhugh–Nagumo system

In this paper, we study the noise-induced temporal and spatial coherence in a spatially extended Fitzhugh–Nagumo (FHN) neuronal system. The system is modeled by a randomly connected small-world network, in which an -stable noise is imposed on the first neuron. We focus on the impact of additive temporally and spatially uncorrelated non-Gaussian -stable noise that determines the interactions amongst coupled FHN neurons. Using numerical simulations, we find that the system presents a typical coherence characteristic on temporal scale, while a decoherence characteristic on spatial scale. Then, the temporal and spatial coherence under the influences of -stable noises are studied, respectively. We find that has a significant impact on the temporal coherence but little significance on the spatial coherence. It shows that in contrast to Gaussian noise, -stable noises lead to poor temporal coherence.

Parameter estimation for non-stationary reflected Ornstein–Uhlenbeck processes driven by [formula omitted]-stable noises

This paper is concerned with the parameter estimation problem for non-stationary reflected Ornstein–Uhlenbeck processes driven by stable noises by using the trajectory fitting method combined with the weighted least squares technique. Under some regularity conditions, we obtain the consistency and the rate of convergence of the proposed estimator of the drift rate. The asymptotic stability property of the estimator in our setting is also proved.

Nonparametric estimation of the trend for stochastic differential equations driven by small [formula omitted]-stable noises

The present paper deals with the problem of nonparametric estimation of the trend for stochastic differential equations driven by small α-stable noises based on continuous-time observation. Under some certain conditions, we derive the uniform consistency and the rate of convergence of the nonparametric estimator. Besides, the asymptotic distribution of the estimator in our setting is shown to be a stable distribution.

Metaheuristic optimization-based identification of fractional-order systems under stable distribution noises

This research investigates the identification problem of fractional-order chaotic systems under stable distribution noises. A powerful metaheuristic optimization method called composite differential evolution is used for the identification of the fractional-order Lorenz and Chen systems in the noisy environment, where the structure, parameters, orders and initial values of the systems are all unknown. The identification accuracy is examined when the noise follows the three special cases of stable distributions, i.e., Gaussian, Cauchy and Lévy distributions. In addition, the impact of the four parameters of stable distributions on the identification accuracy is discussed. The experimental results show that the identification error becomes larger when the noise switches from Gaussian to Cauchy and Lévy distributions. The results also turn out that the location of the stable distribution noise plays the most substantial role in the identification accuracy.

Irreducibility of stochastic real Ginzburg–Landau equation driven by $\alpha$-stable noises and applications

We establish the irreducibility of stochastic real Ginzburg–Landau equation with $\alpha$-stable noises by a maximal inequality and solving a control problem. As applications, we prove that the system converges to its equilibrium measure with exponential rate under a topology stronger than total variation and obeys the moderate deviation principle by constructing some Lyapunov test functions.

Stochastic PDEs with heavy-tailed noise

We analyze the nonlinear stochastic heat equation driven by heavy-tailed noise on unbounded domains and in arbitrary dimension. The existence of a solution is proved even if the noise only has moments up to an order strictly smaller than its Blumenthal–Getoor index. In particular, this includes all stable noises with index α<1+2/d. Although we cannot show uniqueness, the constructed solution is natural in the sense that it is the limit of the solutions to approximative equations obtained by truncating the big jumps of the noise or by restricting its support to a compact set in space. Under growth conditions on the nonlinear term we can further derive moment estimates of the solution, uniformly in space. Finally, the techniques are shown to apply to Volterra equations with kernels bounded by generalized Gaussian densities. This includes, for instance, a large class of uniformly parabolic stochastic PDEs.

Two-time-scale stochastic partial differential equations driven by $\alpha $-stable noises: Averaging principles

This paper focuses on stochastic partial differential equations (SPDEs) under two-time-scale formulation. Distinct from the work in the existing literature, the systems are driven by $\alpha$-stable processes with $\alpha \in(1,2)$. In addition, the SPDEs are either modulated by a continuous-time Markov chain with a finite state space or have an addition fast jump component. The inclusion of the Markov chain is for the needs of treating random environment, whereas the addition of the fast jump process enables the consideration of discontinuity in the sample paths of the fast processes. Assuming either a fast changing Markov switching or an additional fast-varying jump process, this work aims to obtain the averaging principles for such systems. There are several distinct difficulties. First, the noise is not square integrable. Second, in our setup, for the underlying SPDE, there is only a unique mild solution and as a result, there is only mild It\^{o}'s formula that can be used. Moreover, another new aspect is the addition of the fast regime switching and the addition of the fast varying jump processes in the formulation, which enlarges the applicability of the underlying systems. To overcome these difficulties, a semigroup approach is taken. Under suitable conditions, it is proved that the $p$th moment convergence takes place with $p\in(1,\alpha )$, which is stronger than the usual weak convergence approaches.

Maximum likelihood type estimation for discretely observed CIR model with small [formula omitted]-stable noises

A maximum likelihood type estimation of the drift and volatility coefficient parameters in the CIR type model driven by α-stable noises is studied when the dispersion parameter ε→0 and the discrete observations frequency n→∞ simultaneously.

Least squares estimators for stochastic differential equations driven by small Lévy noises

We study parameter estimation for discretely observed stochastic differential equations driven by small Lévy noises. We do not impose Lipschitz condition on the dispersion coefficient function σ and any moment condition on the driving Lévy process, which greatly enhances the applicability of our results to many practical models. Under certain regularity conditions on the drift and dispersion functions, we obtain consistency and rate of convergence of the least squares estimator (LSE) of parameter when ε→0 and n→∞ simultaneously. We present some simulation study on a two-factor financial model driven by stable noises.

Escape process in systems characterized by stable noises and position-dependent resting times.

Stochastic systems characterized by a random driving in a form of the general stable noise are considered. The particle experiences long rests due to the traps the density of which is position dependent and obeys a power-law form attributed to the underlying self-similar structure. Both the one- and two-dimensional cases are analyzed. The random walk description involves a position-dependent waiting time distribution. On the other hand, the stochastic dynamics is formulated in terms of the subordination technique where the random time generator is position dependent. The first passage time problem is addressed by evaluating a first passage time density distribution and an escape rate. The influence of the medium nonhomogeneity on those quantities is demonstrated; moreover, the dependence of the escape rate on the stability index and the memory parameter is evaluated. Results indicate essential differences between the Gaussian case and the case involving Lévy flights.

Improved-Variable-Forgetting-Factor Recursive Algorithm Based on the Logarithmic Cost for Volterra System Identification

Compared with the least-mean-square algorithm, the least mean $p$ th power algorithm shows a better robustness performance against impulsive noises such as the $\alpha$ -stable noises. However, it still exhibits slow convergence rate and high kernel misadjustment. To overcome this drawback, a novel recursive logarithmic least mean $p$ th (RLLMP) algorithm is proposed for the Volterra system identification under $\alpha$ -stable noise environments. Instead of minimizing the $p$ th power, the new algorithm aims to minimize the $p$ th logarithmic cost, which makes it more robust against impulsive interferences. Furthermore, to enhance tracking performance, an improved variable forgetting factor (IVFF) algorithm (IVFF-RLLMP) is proposed, which is based on the robust estimation of outliers. Simulation results are presented to demonstrate the improved performance of the RLLMP and IVFF-RLLMP.