Wiener Process Research Articles

Fractional stochastic heat equation with mixed operator and driven by fractional-type noise

<p>We investigated a novel stochastic fractional partial differential equation (FPDE) characterized by a mixed operator that integrated the standard Laplacian, the fractional Laplacian, and the gradient operator. The equation was driven by a random noise, which admitted a covariance measure structure with respect to the time variable and behaved as a Wiener process in space. Our analysis included establishing the existence of a solution in the general case and deriving an explicit form for its covariance function. Additionally, we delved into a specific case where the noise was modeled as a generalized fractional Brownian motion (gfBm) in time, with a particular emphasis on examining the regularity of the solution's sample paths.</p>

СТОХАСТИЧНЕ МОДЕЛЮВАННЯ ОПТИМАЛЬНИХ ЗВ’ЯЗКІВ МІЖ ТІНЬОВОЮ ТА ЛЕГАЛЬНОЮ ЕКОНОМІКАМИ

The article is devoted to the stochastic modelling of optimal links between the shadow and legal economies using Wiener and Poisson processes. The study focuses on identifying the best management strategies that will help reduce the size of the shadow economy. The paper first considers a deterministic economic and mathematical model of the relationship between the shadow and legal economies, which is presented in the form of differential equations with initial conditions. These differential equations describe the rate of change of real sector workers' savings, the dynamics of owners' savings, the rate of change of the price of aggregate output, etc. The assumptions about the dynamics of cash savings and the price of aggregate output are supplemented by initial conditions. Restrictions on the consumer shares of savings used for consumption of legal and shadow products for employees and entrepreneurs are formed. To build an optimal process, the author formalizes an objective function (objective criterion) that maximizes the average total (integral) savings of employees and entrepreneurs over a certain period of time and uses stochastic sufficient conditions for optimality, where the controls are the shares of savings used to purchase aggregate products of the legal and shadow economies of employees and business owners, and the phase trajectories are the monetary savings of employees and business owners of legal and shadow economies, the price of a unit of aggregate products, and the price of a unit of aggregate products. Thus, the main idea of the process of solving the optimal management problem takes into account both the management process itself (strategies, management methods) and the paths along which the system moves from the initial state to the final result (phase trajectories). The structure of the optimal process of the proposed stochastic model is described. It is established that the optimal controls are deterministic values and do not depend on the coefficients of the Wiener processes in the proposed stochastic dynamic model. It is also worth noting that the specific formulation of this problem of optimal control of the interaction between the shadow and legal economies depends on the context of the study and may vary depending on the tasks and assumptions set.

Approximate controllability of second-order neutral stochastic differential evolution systems with random impulsive effect and state-dependent delay

<p>In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work.</p>

Convergence Analysis of Ensemble Filters for Linear Stochastic Systems with Poisson-Sampled Observations

For continuous-time linear stochastic dynamical systems driven by Wiener processes, we consider the problem of designing ensemble filters when the observation process is randomly time-sampled. We propose a continuous-discrete McKean–Vlasov type diffusion process with additive Gaussian noise in observation model, which is used to describe the evolution of the individual particles in the ensemble. These particles are coupled through the empirical covariance and require less computations for implementation than the optimal ones based on solving Riccati differential equations. Using appropriate analysis tools, we show that the empirical mean and the sample covariance of the ensemble filter converges to the mean and covariance of the optimal filter if the mean sampling rate of the observation process satisfies certain bounds and as the number of particles tends to infinity.

Stochastic solitons of a short-wave intermediate dispersive variable (SIdV) equation

<abstract><p>It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions.</p></abstract>

Existence and controllability of nonlinear evolution equation involving Hilfer fractional derivative with noise and impulsive effect via Rosenblatt process and Poisson jumps

<abstract><p>This manuscript explores a new class of Hilfer fractional stochastic differential system, as driven by the Wiener process and Rosenblatt process through the application of non-instantaneous impulsive effects and Poisson jumps. Existence of a mild solution to the considered system is proved. Sufficient conditions for the controllability of the proposed control system are established. To prove our main results, we utilize fractional calculus, stochastic analysis, semigroup theory, and the Sadovskii fixed point theorem. In addition, to illustrate the theoretical findings, we present an example.</p></abstract>

СТОХАСТИЧНА ДЕСКРИПТОРНА ГРА ПЕРЕСЛІДУВАННЯ

A differential pursuit game in a stochastic descriptor linear system is analyzed. The system dynamics is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the formula of variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct a pursuer control bringing the dynamic vector of the system to a terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transient states in a radio technical filter with random perturbations in the form of white noise. Keywords: stochastic differential algebraic equation, Wiener process, descriptor system, differential game, radio technical filter, white noise.

Learning how to find targets in the micro-world: the case of intermittent active Brownian particles.

Finding the best strategy to minimize the time needed to find a given target is a crucial task both in nature and in reaching decisive technological advances. By considering learning agents able to switch their dynamics between standard and active Brownian motion, here we focus on developing effective target-search behavioral policies for microswimmers navigating a homogeneous environment and searching for targets of unknown position. We exploit projective simulation, a reinforcement learning algorithm, to acquire an efficient stochastic policy represented by the probability of switching the phase, i.e. the navigation mode, in response to the type and the duration of the current phase. Our findings reveal that the target-search efficiency increases with the particle's self-propulsion during the active phase and that, while the optimal duration of the passive case decreases monotonically with the activity, the optimal duration of the active phase displays a non-monotonic behavior.

Degradation Analysis: Modeling and Evaluating for Electric Energy Meters under Multi-stress

The measurement accuracy of electric energy meters (EEMs) is crucial for respecting the fairness of the electricity market and the justice of electric energy settlement. However, the basic error (BE) of the measurement suffers from external stress, especially under extreme environments. It is challenging to evaluate the degradation trend of EEMs under multi-stress. To address this issue, this paper proposes modeling and evaluating methods for precise degradation analysis of EEMs under multi-stress. First, an optimized k-nearest neighbor (OKNN) is proposed to correct the outliers, where the stress-related weight and weighting factor are designed. Then, a varied-bias wiener process with hierarchical Bayesian (VWHB) model is introduced to evaluate and explore the impact of stress on the BE. The multi-stress, as well as the BE, are fused to parameterize this impact. Integrating the OKNN and VWHB, a degradation analysis framework is further proposed to model the data and conduct the degradation analysis. Extensive experiments on field data collected from the typical operating environment lab demonstrated that the proposed degradation analysis framework reveals superior performance than some state-of-art approaches. The root of the mean of the square of errors and the mean of absolute value of errors of OKNN-VWHB are the lowest with 0.0356 and 0.0281, respectively, for all the EEMs from three different companies.

Bivariate and two-phase degradation modeling and reliability analysis with random effects

The paper aims at predicting the remaining useful life of highly reliable and long-life products with multiple and multi-stage characteristics in the degradation process. Considering the unit-to-unit variability among the product units, a new bivariate and two-phase Wiener process model with random effects is established. Schwarz Information Criterion is used to identify the change points of the degradation model, and the analytical expressions of life and remaining useful life are given by the concept of first hitting time. Furthermore, the appropriate Copula function is selected to describe the correlation between the two quality characteristics based on Akaike Information Criterion. A bivariate degradation model is established and the unknown parameters of the model are estimated by Markov Chain Monte Carlo method. Finally, the applicability and effectiveness of the proposed method are verified by the comparative analysis of turbine engine.

Sealing Degradation Modeling and Reliability Analysis Method of Deep‐Water Pipeline Connector Based on the Two‐Phase Nonlinear Wiener Process

The deep‐water pipeline connector (DPC) is a key piece of equipment for the connection of multiple facilities located in different positions of the subsea production system. Establishing a DPC sealing degradation model considering the internal corrosion can provide a basis for its structural design and maintenance. Considering the characteristics of nonlinear and multiphase for the DPC degradation process, a two‐phase nonlinear Wiener process is applied to establish a DPC degradation model. It can reasonably consider the commonly encountered situation that a change point locates among the test times. DPC reliability analysis and lifetime prediction methods are constructed after a maximum likelihood method is employed to estimate the hyperparameters of the established degradation model. Finally, a subsea Christmas tree wellhead connector is taken as an example to verify the reasonability and effectiveness of the method. The results indicate that the proposed method can describe the nonlinear time‐varying properties caused by the DPC internal corrosion more accurate and reasonable than other traditional classic methods, and a more reasonable lifetime assessment result can be obtained.

Energy saving approximation of Wiener process under unilateral constraints

В работе рассматривается энергетически эффективная аппроксимация винеровского процесса при односторонних ограничениях. Показано, что с вероятностью единица на больших интервалах времени минимально необходимая для аппроксимации энергия логарифмически зависит от длины интервала. Построена адаптивная стратегия аппроксимации, оптимальная в классе диффузионных стратегий, также дающая логарифмический порядок расхода энергии.

Multi-Performance Degradation System Reliability Analysis with Varying Failure Threshold Based on Copulas

This paper investigated reliability modeling for systems subject to dependent competing risks considering that variation of the failure threshold is not considered in most studies on competing failure reliability. Firstly, the variation of degradation quantity under shocks was analyzed, and the variation of the threshold was considered on this basis. Secondly, the cumulative degradation under the influence of the random shock process was analyzed. The attractive property of Copula functions is symmetry. Then, a linear Wiener process model was applied to model performance degradation failure, and a multi-performance degradation correlated-competition model based on a Copula function was constructed, which considered the correlated competition between multi-performance degradation failures. Lastly, a micromotor system was used to analyze the applicability of the proposed model for bivariate instances, demonstrating the rationality and effectiveness of the proposed model.

A brief review on stability investigations of numerical methods for systems of stochastic differential equations

<abstract><p>A modest review on stability investigations of numerical methods for systems of Itô-interpreted stochastic differential equations (SDEs) driven by $ m $-dimensional Wiener processes $ W = (W^1, W^2, ..., W^m) $ is presented in $ \mathbb{R}^d $. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of $ \mathbb{R}^{d \times d} $, instead of classic stability functions with values in $ \mathbb{C}^1 $, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with state-independent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size $ h $ is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions $ d \ge 1 $.</p></abstract>

Accelerated Degradation Reliability Modeling Based on Multiple Uncertainty Wiener Processes

Numerous scholars' studies usually adopt the Wiener process to establish industrial robot performance degradation models and randomize the drift coefficients to characterize the differences in individual degradation rates, but they do not take into account the influence of measurement errors on the accuracy of the models. Aiming at the above problems, combining the Wiener process and the accelerated degradation model, the accelerated degradation reliability model of industrial robots based on the multi-uncertainty Wiener process is established, and by adding the factor of measurement error, the model is able to more accurately characterize the influence of external factors on the degradation process of industrial robots. Bayesian theory, Markov chain Monte Carlo method and Bootstrap method are used to estimate the unknown parameters in the model. Finally, the accuracy of the proposed parameter estimation method and model is verified by simulating the industrial robot performance degradation data with Monte Carlo method.

Functional degradation reliability analysis for non-uniform wear of multi-rotating joints of mechanical structures

Failure of the aircraft landing gear lock mechanism can result in the inability of the aircraft to take off and land properly, and its failure can cause critical results. The aircraft lock mechanism is characterized by multiple joints. Based on the performance degradation phenomenon of the lock mechanism, a reliability analysis method of the lock mechanism with non-uniform wear of multiple joints is proposed, which takes into account the effect of non-uniform wear of multiple joints on the motion accuracy of the motion mechanism, and investigates the reliability analysis of the degradation of the angle of the lock hook and the calculation of the whole lifetime cycle. Firstly, the lock mechanism lock hook angle degradation eigenvolume model is established according to the functional principle of the lock mechanism. Secondly, the lock hook degradation angle is obtained by establishing the kinetic equations to calculate the non-uniform wear of each joint separately. Finally, the degradation path is fitted by the Wiener process to carry out the reliability analysis and the whole lifetime calculations. The results show that the proposed method is capable of providing maintenance strategies to meet the reliability requirements for the performance degradation of multi-joint kinematic mechanisms of mechanical systems.

On the Expected Discounted Penalty Function Using Physics-Informed Neural Network

We study the expected discounted penalty at ruin under a stochastic discount rate for the compound Poisson risk model with a threshold dividend strategy. The discount rate is modeled by a Poisson process and a standard Brownian motion. By applying the differentiation method and total expectation formula, we obtain an integrodifferential equation for the expected discounted penalty function. From this integrodifferential equation, a renewal equation and an asymptotic formula satisfied by the expected discounted penalty function are derived. In order to solve the integrodifferential equation, we use a physics-informed neural network (PINN) for the first time in risk theory and obtain the numerical solutions of the expected discounted penalty function in some special cases of the penalty at ruin.

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

Remaining Useful Life Prediction for Two-Phase Nonlinear Degrading Systems with Three-Source Variability.

Recently, the estimation of remaining useful life (RUL) for two-phase nonlinear degrading devices has shown rising momentum for ensuring their safe and reliable operation. The degradation processes of such systems are influenced by the temporal variability, unit-to-unit variability, and measurement variability jointly. However, current studies only consider these three sources of variability partially. To this end, this paper presents a two-phase nonlinear degradation model with three-source variability based on the nonlinear Wiener process. Then, the approximate analytical solution of the RUL with three-source variability is derived under the concept of the first passage time (FPT). For better implementation, the offline model parameter estimation is conducted by the maximum likelihood estimation (MLE), and the Bayesian rule in conjunction with the Kalman filtering (KF) algorithm are utilized for the online model updating. Finally, the effectiveness of the proposed approach is validated through a numerical example and a practical case study of the capacitor degradation data. The results show that it is necessary to incorporate three-source variability simultaneously into the RUL prediction of the two-phase nonlinear degrading systems.

Effects of Wiener process on analytical wave solutions for (3+1) dimensional nonlinear Schrödinger equation using modified extended mapping method

In this study, we investigate the soliton solutions for stochastic (3+1) dimensional nonlinear Schrödinger equation (NLSE). Our Study mainly depends on applying the modified extended mapping method to construct various and novel stochastic solutions for the proposed model. These solutions including (dark, bright, and singular) solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, and Jacobi elliptic function (JEF) solutions are raised. For a variety of nonlinear stochastic partial differential equations, this method offers a practical and effective method for determining exact stochastic solutions. The stochastic noise effects forcing the wave to behave differently from deterministic cases prognosticated. To illustrate the physical characteristics of the established stochastic solutions, some selected solutions are presented graphically. We used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs.