Wiener Process Research Articles

SPDE bridges with observation noise and their spatial approximation

This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.

On the Convergence of Random Fourier-Jacobi Series of Continuous functions

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $\gamma,\delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.

A two-stage Gaussian process regression model for remaining useful prediction of bearings

Bearing is one of the most important supporting components in mechanical equipment and its health status has a significant impact on the overall performance of equipment. The remaining useful life (RUL) prediction of bearings is critical in adopting a condition-based maintenance strategy to ensure reliable equipment operation. To accurately predict the RUL of bearings, this paper proposes a two-stage Gaussian process regression (GPR) model, which combines the flexibility of the Gaussian process and the physical mechanism of the Wiener process. Compared with the conventional GPR model, the proposed model can reasonably adapt to the statistical characteristics of bearings degradation and provide more stable predictions. In addition, the paper proposes a new degradation detection approach based on the Euclidean distance to distinguish the two stages of the bearing service life cycle, which considers the global characteristics of bearing degradation and can accurately detect the beginning point of bearing degradation. The experimental results show that the proposed two-stage GPR model can help to improve the precision and accuracy of degradation path tracking and RUL prediction.

A survey of mean-square destabilization of multidimensional linear stochastic differential systems with non-normal drift

Mean-square stability analysis of linear stochastic differential systems obtained perturbing ordinary systems by linear terms driven by independent Wiener processes is investigated. The so obtained stochastic regions are contractions of the asymptotic stability domain of the linear ordinary system. In this work, the mean-square stability regions exact shape is provided by means of necessary and sufficient conditions in terms of the eigenvalues of the drift and the intensities of the noises. Special attention is paid to how different structures of the perturbation affect the mean-square stability of systems with non-normal drifts. In each case, the obtained explicit stability condition reveals the role played by the parameter that controls the non-normality.

Stochastic modeling of fatigue crack growth for bolt holes in turbine disc

The stochastic fatigue crack growth (FCG) modeling method is established for bolt holes in the turbine disc. Through analyzing the FCG experimental of standard compact tension (CT) specimens, a Wiener process is firstly established based on the failure mechanism to describe the stochastic FCG process, and then extended to the component-like and component level. The accuracy of the established model is verified by the experiments of bolt hole plate (BHP) specimens. Finally, the reliability analysis framework is constructed and verified for the turbine disc with multiple cracks based on the Copula function.

Predefined-time synchronization of coupled neural networks with switching parameters and disturbed by Brownian motion

This article focuses on predefined time synchronization problem for a class of signal switching neural networks with time-varying delays. In the network models, we not only consider the coupling characteristics in the following networks, but also consider the disturbance with standard Brownian motion. In the design of the controller, the control gain is designed as 1ɛ+Tp−t (t∈[T0,Tp), ɛ is an optional smaller positive number), which avoids the infinite gain (the control gain is designed as 1Tp−t in other reference). In order to get the predefined time control law, a power function is multiplied to the Lyapunov functional, from which it can get an exponential upper bound function via the derivative and mathematical expectation operation. Utilizing the martingale theory and the method of Laplace matrix, some novel predefined time synchronization criteria are obtained for the leader-following neural networks, meanwhile the following networks can maintain the leader network after achieved synchronization. Based on the special network of the main system, five corollaries separately develop the predefined time synchronization results from different perspectives. An example with some simulation figures and computing results fully exhibits the effectiveness of the achieved synchronization scheme. In this case, although the error signal is disturbed by Brownian motion, the trace signal can still stably converge to zero by this control scheme, meanwhile the predefined-time control effect is achieved.

A Multivariate Stochastic Degradation Model for Dependent Performance Characteristics

Simultaneous degradation of multiple dependent performance characteristics (PCs) is a common phenomenon for industrial products. The associated degradation modeling is of practical importance yet challenging. The dependence of the PCs can usually be attributed to two sources, one being the overall system health status and the other the common operating environments. Based on the observation, this study proposes a parsimonious multivariate Wiener process model whose number of parameters increases linearly with the dimension. We introduce a common stochastic time scale shared by all the PCs to model the dependence from the dynamic operating environment. Conditional on the time scale, the degradation of each PC is modeled as the sum of two independent Wiener processes, where one represents the common effects shared by all the PCs, and the other represents degradation caused by randomness unique to this PC. An EM algorithm is developed for model parameter estimation, and extensive simulations are implemented to validate the proposed model and the algorithms. For efficient reliability evaluation under a multivariate degradation model, including the proposed one, a bridge sampling-based algorithm is further developed. The applicability and the advantages of the proposed methods are demonstrated using a multivariate degradation dataset of a coating material. Supplementary materials for this article are available online.

Reliability inference and remaining useful life prediction for the doubly accelerated degradation model based on Wiener process

<abstract><p>Degradation data are an important source of products' reliability information. Though stochastic degradation models have been widely used for fitting degradation data, there is a lack of efficient and accurate methods to get their confidence intervals, especially in small sample cases. In this paper, based on the Wiener process, a doubly accelerated degradation test model is proposed, in which both the drift and diffusion parameters are affected by the stress level. The point estimates of model parameters are derived by constructing a regression model. Furthermore, based on the point estimates of model parameters, the interval estimation procedures are developed for the proposed model by constructing generalized pivotal quantities. First, the generalized confidence intervals of model parameters are developed. Second, based on the generalized pivotal quantities of model parameters, using the substitution method the generalized confidence intervals for some interesting quantities, such as the degradation rate $ \mu_0 $, the diffusion parameter $ \sigma_0^2 $, the reliability function $ R(t_0) $ and the mean lifetime $ E(T) $, are obtained. In addition, the generalized prediction intervals for degradation amount $ X_0(t) $ and remaining useful life at the normal use stress level are also developed. Extensive simulations are conducted to investigate the performances of the proposed generalized confidence intervals in terms of coverage percentage and average interval length. Finally, a real data set is given to illustrate the proposed model.</p></abstract>

On stochastic elliptic equations driven by Wiener process with non-local condition

In this paper, a problem for stochastic elliptic equations containing Wiener process and non-local condition is considered. Initially, existence results are constructed under two different hypotheses for the input data. Next, the instability behavior of the mild solution is presented, which shows that the solution does not depend continuously on the given data. After that, a regularized solution is established and its convergence rate is described. Lastly, an illustrated example is shown to test the proposed method. Our article was inspired by recent papers by Tomás Caraballo and his colleagues on the ill-posed problem for stochastic PDEs.

Sticky Feller diffusions

We consider a Feller branching diffusion process X with drift c having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter 1∕μ∈(0,∞). We show that (i) the process X can be characterised as a unique weak solution to the SDE system dXt=(bXt+c)I(Xt>0)dt+ 2aXtdBt I(Xt=0)dt= 1μdℓt0(X) where b∈R and 0<c<a are given and fixed, B is a standard Brownian motion, and ℓ0(X) is a diffusion local time process of X at 0, and (ii) the transition density function of X can be expressed in the closed form by means of a convolution integral involving a new special function and a modified Bessel function of the second kind. The new special function embodies the stickiness of X entirely and reduces to the Mittag-Leffler function when b=0. We determine a (sticky) boundary condition at zero that characterises the transition density function of X as a unique solution to the Kolmogorov forward/backward equation of X. Letting μ↓0 (absorption) and μ↑∞ (instantaneous reflection) the closed-form expression for the transition density function of X reduces to the ones found by Feller [6] and Molchanov [14] respectively. The results derived for sticky Feller diffusions translate over to yield closed-form expressions for the transition density functions of (a) sticky Cox-Ingersoll-Ross processes and (b) sticky reflecting Vasicek processes that can be used to model slowly reflecting interest rates.

Asymptotically normal estimation of parameters of mixed fractional Brownian motion

We investigate the mixed fractional Brownian motion of the form Xt = σ Wt + κ BtH, driven by a standard Brownian motion W and a fractional Brownian motion BH with Hurst parameter H. We consider strongly consistent estimators of unknown model parameters (H, κ, σ) based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for H ∈ (0, 1/2) ∪ (1/2, 3/4).

Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes.

We analyze the transition probability density functions in the presence of a zero-flux condition in the zero-state and their asymptotic behaviors for the Wiener, Ornstein Uhlenbeck and Feller diffusion processes. Particular attention is paid to the time-inhomogeneous proportional cases and to the time-homogeneous cases. A detailed study of the moments of first-passage time and of their asymptotic behaviors is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Specific applications of the results to queueing systems are provided.

Simultaneous Multiplicative and Additive Actuator Faults Estimation‐Based Sliding Mode FTC for a Class of Uncertain Nonlinear System

In this article, an adaptive sliding mode fault tolerant control (FTC) is improved in the case of uncertain nonlinear system which is affected by both multiplicative and additive faults in actuator. Especially, when the nonlinear system is modeled by Takagi–Sugeno (T‐S) fuzzy system with local nonlinear model. The main contribution of this paper is developing a model of multiplicative faults, which offers a more realistic dynamic evolution of the actuator degradation. The degradation process is modeled by Wiener process and estimated by the maximum likelihood estimation (MLE). Sliding mode observer (SMO) is conceived to realize the additive actuator faults using convex multiobjective optimization. On these bases, the estimated multiplicative and additive actuator faults are used to design the adaptive sliding mode controller (SMC). Finally, the proposed fault‐tolerant control scheme is demonstrated by the results of inverted pendulum system simulation.

Parameter estimation in mixed fractional stochastic heat equation

The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

Conditioned Wiener processes as nonlinearities: A rigorous probabilistic analysis of dynamics

We study a Wiener process that is conditioned to pass through a finite set of points and consider the dynamics generated by iterating a sample path from this process. Using topological techniques we are able to characterize the global dynamics and deduce the existence, structure and approximate location of invariant sets. Most importantly, we compute the probability that this characterization is correct. This work is probabilistic in nature and intended to provide a theoretical foundation for the statistical analysis of dynamical systems which can only be queried via finite samples.

Conditional wiener integral associated with Gaussian processes and applications

Let C0[0,T] denote the one-parameter Wiener space and let C?0[0,T] be the Cameron-Martin space in C0[0,T]. Given a function k in C?0[0,T], define a stochastic process Zk : C0[0,T] ? [0, T] ? R by Zk(x, t) = R t 0 Dk(s)dx(s), where Dk ? d/dt k. Let a random vector XG,k : C0[0,T] ? Rn be given by XG,k(x) = ((g1,Zk(x,?))~,..., (gn,Zk(x,?))~), where G = {g1, ..., gn} is an orthonormal set with respect to the weighted inner product induced by the function k on the space C?0[0,T], and (g,Zk(x,?))~ denotes the Paley-Wiener-Zygmund stochastic integral. In this paper, using the reproducing kernel property of the Cameron-Martin space, we establish a very general evaluation formula for expressing conditional generalized Wiener integrals, E(F(Zk(x,?)) |XG,k(x) = ??), associated with the Gaussian processes Zk. As an application, we establish a translation theorem for the conditional Wiener integral and then use it to obtain various conditional Wiener integration formulas on C0[0,T].

On two-dimensional extensions of Bougerol’s identity in law

Let B={Bt}t≥0 be a one-dimensional standard Brownian motion and denote by At,t≥0, the quadratic variation of semimartingale eBt,t≥0. The celebrated Bougerol’s identity in law (1983) asserts that, if β={βt}t≥0 is another Brownian motion independent of B, then βAt has the same law as sinhBt for every fixed t>0. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving, as the second coordinates, the local times of B and β at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.

Monte Carlo study of uncertainty propagation in Euler rotational kinematic equation

We perform a Monte Carlo simulation to study the propagation of uncertainty in Euler rotational kinematic equation. The rate gyro data from an IMU sensor are inserted into the Euler rotational kinematic differential equation to calculate the sensor orientation, which is specified by its Euler angles. The uncertainty of sensor orientation is determined by simulating the variation of rate gyro input due to its uncertainty obtained from the calibration process. We find that the uncertainties of the Euler angles grow with time. Their values are proportional to the square root of time elapse. We find also that the uncertainties are linearly proportional to the uncertainties of rate gyro data and the square root of sampling time. These results interestingly resemble the behaviour of the Wiener process, with the step size of random walk depends on the time step. We can use these results as a criterion to choose a suitable IMU due to the characteristic of the integrated rate gyro sensor, especially in the application like the vehicle tracking or posture warning for elderly/disable people, where the precise sensor orientation is needed for a long period of operation.

Elementary processes for Itô Integral against cylindrical Wiener process

Elementary processes for Itô Integral against cylindrical Wiener process