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Accurate remaining useful life (RUL) of batteries plays an imperative role in ensuring safe operations and avoiding catastrophic accidents. However, in practice, complicated working conditions bring challenges to accurate battery prognostics. In this study, an accelerated stress factors based nonlinear Wiener process model is proposed to enrich inadequate battery prognostic works at various operating conditions. To realize online individual battery prognostics, once a new measurement is available, the parameters of a state-space model constructed by the proposed model are posteriorly updated. Then, based on the Peukert law and the Arrhenius equation, two specific accelerated stress-relevant drift functions and their associated degradation models at different discharge rates and temperatures are respectively designed. Subsequently, RUL predictions are conducted using the proposed method. RUL predictions at different discharge rates and different temperatures demonstrate the accuracy and robustness of the proposed prognostic models. According to some general prognostic metrics, the proposed method is proved to be superior to four existing RUL prediction approaches.

We consider a Bessel process X of dimension δ ∈ ( 0 , 2 ) 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 through its square Y = X 2 as a unique weak solution to the SDE system d Y t = δ I ( Y t > 0 ) d t + 2 Y t d B t I ( Y t = 0 ) d t = 1 2 μ d l t 0 ( Y ) where B is a standard Brownian motion and l 0 ( Y ) is a diffusion local time process of Y at 0, and (ii) the transition density function of X can be expressed in the closed form as a convolution integral involving a Mittag-Leffler function and a modified Bessel function of the second kind. Appearance of the Mittag-Leffler function is novel in this context. We determine a (sticky) boundary condition at zero that characterises the transition density function of X as a unique solution to the Kolmogorov backward/forward equation of X . We also show that the convolution integral can be characterised as a unique solution to the generalised Abel equation of the second kind. 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 (1951) and Molchanov (1967) respectively.

This article proposes a novel modeling method for the stochastic nonlinear degradation process by using the relevance vector machine (RVM), which can describe the nonlinearity of degradation process more flexibly and accurately. Compared with the existing methods, where degradation processes are modeled as the Wiener process with a nonlinear drift function formulized as the power law or exponential law, this kind of modeling method can characterize degradation processes with more nonlinear behavior. Instead of modeling the drift coefficient of the Wiener process directly, the weighted combination of basis functions is utilized to express the increment of the Wiener process and the parameters are calculated by a sparse Bayesian learning algorithm. Based on the proposed model, a numerical approximation formula for the probability density function (PDF) of the remaining useful life (RUL) is derived. Finally, comparison studies, including a numerical simulation and a practical case, are provided to demonstrate the effectiveness and the accuracy of the proposed methods for RUL estimation.

Given n independent random vectors with common density f on R d , we study the weak convergence of three empirical-measure based estimators of the convex λ-level set L λ of f , namely the excess mass set, the minimum volume set and the maximum probability set, all selected from a class of convex sets A that contains L λ. Since these set-valued estimators approach L λ , even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of L λ and each of the three estimators, at rate n −1/3. It turns out that the minimum volume set and the maximum probability set estimators are asymptotically indistinguishable, whereas the excess mass set estimator exhibits "richer" limit behavior. Arguments rely on the boundary local empirical process, its cylinder representation, dimension-free concentration around the boundary of L λ , and the set-valued argmax of a drifted Wiener process. MSC 2010 subject classifications: Primary 62G05, 62G20; secondary 60F05, 60F17.

Given a Wiener process with unknown and unobservable drift, we try to estimate this drift as effectively but also as quickly as possible, in the presence of a quadratic penalty for the estimation error and of a fixed, positive cost per unit of observation time. In a Bayesian framework, where the unobservable drift is assumed to have a known “prior” distribution, this question reduces to choosing judiciously a stopping time for an appropriate diffusion process in natural scale. We establish structural properties of the solution for the corresponding problem of optimal stopping. In particular, we show that, regardless of the prior distribution, the continuation region is monotonically shrinking in time. Moreover, we provide conditions on the prior distribution that guarantee a one-sided stopping region. Lastly, some concrete prior distributions are studied to illustrate the theoretical results.

The degradation data of highly reliable products are usually analyzed by stochastic process models, such as Wiener process, gamma process and inverse Gaussian process models. If such a specific degradation model is wrongly assumed, then poor analysis results of reliability assessment would be obtained. Therefore, a class of exponential-dispersion processes, named Tweedie exponential-dispersion process (TEDP), is proposed to describe the products’ degradation paths. The TEDP model which comprises the aforementioned stochastic processes as its special cases, is more flexible and applicable for degradation modeling. Considering the nonlinear characteristics of degradation paths and the unit-to-unit variability among the product units, random-effect models are established based on the nonlinear TEDP models with random drift and dispersion parameters. To improve the mathematical tractability of these models, the variational inference, expectation maximization algorithm and differential evolution algorithm are used to estimate the unknown model parameters. Furthermore, two nonlinear TEDP models with accelerated factors and random effects are developed for accelerated degradation analysis. Finally, a simulation study and three real applications are presented to show the effectiveness and superiority of the proposed models and methods.

Aviation and aerospace system are typical Phased-Mission Systems (PMSs) featured with varying configuration, phased load condition and cross phase failure correlation. The coupling effect of Functional Dependency (FDEP) and Physical Dependency (PDEP) has a unique influence on the failure behavior of PMS. In this article, the coupling effect is analyzed, and the degradation of components is modeled with the positive drift wiener process, in which the drift coefficient is related to environmental conditions and operation stress. Finally, failure behavior and system reliability are simulated. An avionics controller is studied as a case, with the degradation time distribution model and simulation algorithm, the coupling effect and dynamical system reliability can be achieved. Results show that this modeling method can describe the coupling effects of FDEP and PDEP and their influence on the failure behavior and reliability of the PMS system.

We present an exact distribution for the bounded range of a Wiener process by using the truncation method of the probability distributions. Rather than deriving some statistical properties of this ...

Given a probability space (Ω, \( \mathcal{F} \), P), where Ω denotes a non-empty set, \( \mathcal{F} \) a σ-algebra and P a probability measure on Ω: A random variable (r.v.) is a function X : Ω → ℝ such that {ω ∈ Ω : X (ω) ∈ A} ∈ F for any Borel set A in ℝ. A stochastic process is a family (X t ) t≥0 of random variables defined on (Ω, \( \mathcal{F} \), P). The stochastic process (X t ) t≥0 is said to be a discrete-time stochastic process if t takes values in ℕ and a continuous-time stochastic process if t takes values in ℝ+. A filtration on Ω is a family \( \left( {\mathcal{F}_t } \right)_{t \geqslant 0} \) of σ-algebras on Ω such that \( \mathcal{F}_u \subseteq \mathcal{F}_v \) for any u ≤ v.

Most systems can fail in multiple ways, and the failure modes are usually positively correlated. This phenomenon complicates the reliability analysis and makes the corresponding maintenance planning challenging. This article proposes a condition-based maintenance policy for systems that are subject to both degradation-induced soft failure and sudden hard failure, where a higher degradation level leads to a higher hazard rate of the hard failure. The Wiener process is adopted for the degradation process, and the Weibull model is used to describe the baseline hazard rate of the hard failure. The degradation level is then treated as a time-varying covariate that affects the hazard rate of the hard failure, and the closed-form of the reliability function is derived by using the Brownian bridge theory. An inspection/replacement maintenance policy is employed, and the long-run cost rate is formulated based on the semiregenerative property of the system state. The optimal inspection interval and the preventive replacement threshold are then jointly determined by minimizing the long-run cost rate. A numerical study on a hydraulic sliding spool system is conducted to validate the derived reliability function and the maintenance policy.

For safety-critical industrial systems, standby redundancy is widely applied to guarantee safety requirements. With the advancement of sensing technology, real-time system reliability evaluation can be executed to prevent failures, for which the load-sharing characteristic has to be considered because multiple components jointly accomplish the mission. In this article, a novel methodology is proposed to online assess the reliability and optimize the mode transfer scheduling of the degrading standby system considering the load-sharing characteristic. First, the stepwise drifted Wiener process is invoked to characterize the component degradation process. Then, to deal with the process and measurement noise, the monitored degradation data are processed online to update the component degradation state and rate, based on the particle filter. Besides, a Monte Carlo method involving the closed Newton–Cotes quadrature rule is employed to quantify the real-time system reliability. Finally, the mode transfer scheduling of standby components is optimized online to maximize the system reliability at mission time. A numerical example and a case study referring to the aircraft hydraulic system are presented to demonstrate the effectiveness of the developed methodology.

In this paper, we construct the two-parameter semigroup of operators associated with a certain one-dimensional inhomogeneous diffusion process and study its properties. We are interested in the process on the real line which can be described as follows. At the interior points of the half-lines separated by a point, the position of which depends on the time variable, this process coincides with the Wiener process given there and its behavior on the common boundary of these half-lines is determined by a kind of the conjugation condition of Feller-Wentzell's type. The conjugation condition we consider is local and contains only the first-order derivatives of the unknown function with respect to each of its variables. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding conjugation problem for a second order linear parabolic equation to which the above problem is reduced. Its classical solvability is obtained by the boundary integral equations method under the assumption that the initial function is bounded and continuous on the whole real line, the parameters characterizing the Feller-Wentzell conjugation condition are continuous functions of the time variable, and the curve defining the common boundary of the domains is determined by the function which is continuously differentiable and its derivative satisfies the Hölder condition with exponent less than $1/2$.

The performance of photovoltaic (PV) modules gradually declines as the operational time continues due to the impacts of workload and environmental factors. This performance degradation process is dynamic and random. Hence, two stochastic process models, exponential dispersion process with fixed effect and with random effect are proposed to describe this degradation law. With these two models, the procedure for evaluating the reliability of PV modules is illustrated based on a practical case. The proposed model is demonstrated to be rationality and applicability for the PV modules through three common information criteria and Kolmogorov–Smirnov test. Due to the exponential dispersion process includes some common stochastic process, e.g. Wiener process, Inverse Gaussian process and Gamma process, which makes the proposed models have a wide range of applications. The proposed models can provide references for the warranty life, maintenance of PV modules.

This article is concerned with the identification of a Wiener model, which is a widely used block-oriented nonlinear model. The identification of the Wiener model in the presence of outliers and process noises has not been well studied to date. We propose a graphical model to describe the Wiener process, where the outliers are modeled by Student's $t\text{-}$ distribution. Due to the negative effects of the nonlinearity and latent variables on parameter estimation, the model is estimated using a variational Bayesian approach. We propose an effective method to approximate the posterior distributions of the latent variables. As a result, a new method for Wiener model identification is proposed. Compared with the straightforward maximum-likelihood estimation and the classical prediction error method, the proposed method is more robust and provides more accurate parameter estimates. The proposed method is used to identify the temperature dynamic of the phase-change cooling module of the high-performance chip.

In practice, degradation modeling and prognostics for a product working in field is of importance for condition-based maintenance, meanwhile a challenging work due to the uncertainty in degradation and the time-varying operating conditions. This study investigates a degradation modeling framework for field working product and its applications by incorporating multiple data sources to associate the operating condition and reduce the uncertainty. The Wiener process is adopted to model the degradation process with nonlinearity, unit-to-unit variation, and condition covariates, which is delineated by state space modeling. Meanwhile, the acceleration factor, which establishes the relationship between operating condition and degradation rate, is integrated to adapt the drift parameter. A recursive filtering algorithm based on Bayesian theorem is employed for online updating of drift. Finally, a simulation study and an application on degradation modeling of turbofan engine are given to demonstrate the feasibility and validity of the proposed model.

In this paper, a novel remaining useful life prediction approach considering fault effects is proposed. The Wiener process is used to construct the degradation process of single performance characteristic with the fault effects. The first passage time based remaining useful life distribution is calculated by assuming fault occurrence moment is a random variable and follows a certain distribution. Expectation maximization algorithm is employed to estimate model parameters, where the fault occurrence moment is considered as a missing data. Finally, a Copula function is used to describe the dependence between the multiple performance characteristics and derive joint remaining useful life (RUL) distribution of product with the fault effects. The effectiveness of the proposed approach is verified by the experiments of turbofan engines.

This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion \begin{align*} \left( ^{\mathcal C} \partial^\gamma_t + A \right) u(t) = f(t) + B(t) \dot{W}(t), \quad t\in J:=(0,T), \end{align*} where $^{\mathcal C} \partial^\gamma_t$ is the conformable derivative, $\gamma \in (\frac{1}{2},1].$ Under some flexible assumptions on $f,B$ and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces $L^r(J;L^2(\Omega,\dot{H}^\sigma))$ and $C^\alpha(\overline{J};L^2(\Omega,H))$ separately.

In this paper, we study the long memory property of two processes based on the Ornstein-Uhlenbeck model. Their are extensions of the Ornstein-Uhlenbeck system for which in the classic version we replace the standard Brownian motion (or other L$$\acute{e}$$ e ´ vy process) by long range dependent processes based on $$\alpha -$$ α - stable distribution. One way of characterizing long- and short-range dependence of second order processes is in terms of autocovariance function. However, for systems with infinite variance the classic measure is not defined, therefore there is a need to consider alternative measures on the basis of which the long range dependence can be recognized. In this paper, we study three alternative measures adequate for $$\alpha -$$ α - stable-based processes. We calculate them for examined processes and indicate their asymptotic behavior. We show that one of the analyzed Ornstein-Uhlenbeck process exhibits long memory property while the second does not. Moreover, we show the ratio of two introduced measures is limited which can be a starting point to introduction of a new estimation method of stability index for analyzed Ornstein-Uhlenbeck processes.

Predicting the life of Ni-Cd battery for electric multiple units (EMU) can not only improve the safety and reliability of battery, but also reduce the operating costs of EMU. For this reason, a life prediction method based on linear Wiener process is proposed, which is suitable for both monotonic and non-monotonic degraded systems with accurate results. Firstly, a unary linear Wiener degradation model is established, and the parameters of the model are estimated by using the expectation-maximization algorithm (EM). With the established model, the remaining useful life (RUL) of Ni-Cd battery and its distribution are obtained. Then based on the unary Wiener process degradation model, the correlation between capacity and energy is analyzed through Copula function to build a binary linear Wiener degradation model, where its parameters are estimated using Markov Chain Monte Carlo (MCMC) method. Finally, according to the binary Wiener process model, the battery RUL and its distribution are acquired. The experimental results show that the binary linear Wiener degradation model based on capacity and energy possesses higher accuracy than the unary linear wiener process degradation model.

In this article, we study stochastic functional linear models (SFLM) driven by an underlying square integrable stochastic process X(t) which is generated by a standard Brownian motion. Utilizing the magnificent Ito integrals and Malliavin calculus, X(t) is expanded into a summation of orthogonal multiple integrals, i.e., Wiener-Ito chaos expansions, which is the counterpart of the Taylor expansion of deterministic functions. Based on the expansion, we show that the fourth moments of linear functionals of underlying stochastic process X(t) are bounded by the square of their second moments when X(t) is a finite linear combination of multiple Ito integrals. Therefore, an optimal minimax convergence rate in mean prediction risk of SFLM is valid if eigenvalues of related linear operators are of order $$k^{-2r}$$ by using results in literature when the underlying process X(t) is a linear combination of multiple Ito integrals. A sufficient and necessary condition of finite fourth moment of random functions of multiple Ito integrals is proved, which is a key condition in methodology and convergence rates of functional linear regressions. Our results show that the optimal minimax convergence rate in mean prediction risk can be applied to the class of linear combination of multiple Ito integrals which are not necessarily Gaussian processes. Moreover, the sufficient and necessary condition of finite fourth moment for multiple Ito integrals can be directly applied to show methodology and convergence rates of functional linear models. Using the theory of stochastic analysis, one may construct a reproducing kernel Hilbert space (RKHS) associated with a square integrable stochastic process to facilitate analysis of functional data.

We investigate the mixed fractional Brownian motion with trend of the form $$X_t = \theta t + \sigma W_t + \kappa B^H_t$$ , driven by a standard Brownian motion W and a fractional Brownian motion $$B^H$$ with Hurst parameter H. We develop and compare two approaches to estimation of four unknown parameters $$\theta $$ , $$\sigma $$ , $$\kappa $$ and H by discrete observations. The first algorithm is more traditional: we estimate $$\sigma $$ , $$\kappa $$ and H using the quadratic variations, while the estimator of $$\theta $$ is obtained as a discretization of a continuous-time estimator of maximum likelihood type. This approach has several limitations, in particular, it assumes that $$H

This paper proposes an adaptive remaining useful life (RUL) estimation method for partially observable degrading products with time-varying random shocks. In the modeling aspect of the proposed method, a shock degradation model is proposed to characterize the degradation process of the product, in which the continuous degradation process is described by the Wiener process, and the random shock process with a time-varying intensity is described by the non-homogeneous compound Poisson process (NHCPP). In the proposed model, to characterize the effect of random shocks on the degradation process, the degradation rate is related to the number of shocks and the magnitude of cumulative shocks. In the estimating aspect, we utilize a strong tracking filtering (STF) algorithm to estimate partially observable degradation states and a two-step expectation conditional maximization (ECM) algorithm to estimate the model parameters. In the prognostic aspect, the approximated analytical RUL distribution under the concept of the first hitting time (FHT) is obtained by incorporating the unit-to-unit variability and the uncertainty of the partially observable degradation state estimation from observations into the RUL estimation. As such, the RUL distribution can be updated according to the latest available degradation observation, thereby realizing the adaptive RUL estimation. Finally, the accuracy and effectiveness of the proposed approach is verified by a numerical example and a practical case study for the furnace wall, which provides higher estimation accuracy and better online capability than existing approaches.

In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.

We consider together the retrospective and the sequential change-point detection in a general class of integer-valued time series. The conditional mean of the process depends on a parameter $$\theta ^*$$ which may change over time. We propose procedures which are based on the Poisson quasi-maximum likelihood estimator of the parameter, and where the updated estimator is computed without the historical observations in the sequential framework. For both the retrospective and the sequential detection, the test statistics converge to some distributions obtained from the standard Brownian motion under the null hypothesis of no change and diverge to infinity under the alternative; that is, these procedures are consistent. Some results of simulations as well as real data application are provided.

Fiber optical gyroscope (FOG) is a highly reliable navigation element, and the degradation trajectories of its two accuracy indexes are monotonic and non-monotonic respectively. In this paper, a flexible accelerated degradation testing (ADT) model is used for analyzing the bivariate dependent degradation process of FOG. The time-varying copulas are employed to consider the dynamic dependency structure between two marginal degradation processes as the Wiener process and the inverse Gaussian process. The statistical inference is implemented by utilizing an inference function for the margins (IFM) approach. It is demonstrated that the proposed method is powerful in modeling the joint distribution with various margins.

For an $${\text {L}}_2$$ -bounded martingale starting at 0 and having final variance $$\sigma ^2$$ , the expected local time at $$a \in \text {R}$$ is at most $$\sqrt{\sigma ^2+a^2}-|a|$$ . This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $$(a-\sqrt{\sigma ^2+a^2},a+\sqrt{\sigma ^2+a^2})$$ . In particular, the maximal expected local time anywhere is at most $$\sigma $$ , and this bound is sharp. Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals have been established by Dubins and Schwarz (Societe Mathematique de France, Asterisque 157(8), 129–145 1988), by Dubins et al. (Ann Probab 37(1), 393–402 2009) and by the authors (2018).

In this paper we propose an objective Bayesian method to analyze the accelerated degradation model based on the new Wiener process stated in Ye, Chen, and Shen (Reliab. Eng. Syst. Safe., 2015, 139:...

International audience; Traditionally, in reliability acceptance sampling plans, the decision to accept or reject a lot is made by performing the life tests of items. However, when the item's deterioration is described by a degradation process, it can be made based on the observed deterioration levels of the items obtained from degradation tests. In this paper, two acceptance sampling plans are developed, based on the observation of the deterioration of the items, accumulated on a given period of time. To model the degradation of the items over time, the Wiener process with positive drift is employed. Algorithms to find the parameters of the proposed sampling plans are suggested. Conditionally on the acceptance in the test, the developed sampling plans are shown to improve the reliability performance of the items in the sense that the lifetimes of the items after the reliability sampling test are stochastically larger than those before the test. Also, we compare the two sampling plans both from a technical and economical points of view.

In this paper, a class of anticipating stochastic differential equation is considered, where the integrand processes are not adapted to the filtration generated by a Wiener process. Using the correspondence between the Skorohod integral and the Ito–Skorohod integral, the equations can be solved by using standard iterative techniques. The existence and uniqueness of strong solutions to these equations are discussed. Such equations with non-adapted, fuzzy and random processes can be applied in financial models.

Degradation is one of the major root causes of system failure. In some applications, the degradation levels are different upon failure, in which the fixed failure threshold assumption commonly adopted in the degradation literature may not hold. This article tackles the difficulty by jointly analyzing the system degradation and the lifetime data, which enables the corresponding remaining useful life (RUL) prediction. We treat the degradation level as a multiplicative time-varying covariate of the system hazard rate, where a random-effects Wiener process is adopted to model the degradation process. The model parameters are estimated under a Bayesian framework, and we also develop a particle filter method to update the estimates when new data are available. This makes the proposed model be able to realize online RUL prediction based on the in-situ system health state signals. Through case studies on lead-acid batteries and digital communication systems, the proposed model is shown to outperform existing methods in terms of the RUL prediction accuracy.

Using an analytic approach, the critical velocity of chaos for an axially moving viscoelastic string under a noisy axial tension was studied in this paper. A Wiener process non-Gaussian bonded noise was selected to model the noisy fluctuations of axial force, and the Melnikov-based criterion was chosen to obtain the boundaries of chaotic behavior and consequently the critical velocity. The latter was obtained in terms of the amplitude, frequency and bandwidth of the noise. The effect of variation of the elastic and the viscous properties of the string on the qualitative and quantitative behaviors was also investigated. It was observed that the unstable area and the critical velocity depend completely on the bandwidth of the noise. Also, to ensure the correctness of the method, some of the results were validated using the corresponding Poincare maps.

We prove a uniform large deviations principle for the law of the solutions to a class of Burgers-type stochastic partial differential equations in any space dimension. The equation has nonlinearities of polynomial growth of any order, the driving noise is a finite dimensional Wiener process, and the proof is based on variational principle methods. We prove the uniform large deviations principle for the law of the solutions in two different topologies. First, in the C([0, T] : Lρ(D)) topology where the uniformity is over Lρ(D)-bounded sets of initial conditions, and secondly in the $C([0,T]\times \bar D)$ topology with uniformity being over bounded subsets in the $C_{0}(\bar D)$ norm.

In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately $1$ for trace class noise and $\frac{1}{2}$ for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided

We consider singular SDEs like \begin{equation} \label{ss} dX_t = b(t, X_t) dt + A X_t dt + \sigma(t) d{L}_t , \;\; t \in [0,T], \;\; X_0 =x \in {\mathbb R}^n, \end{equation} where $A$ is a real $n \times n $ matrix, i.e., $A \in {\mathbb R}^n \otimes {\mathbb R}^n$, $b$ is bounded and Holder continuous, $\sigma : [0,\infty) \to {\mathbb R}^n \otimes {\mathbb R}^d $ is a locally bounded function and $L= ({L}_t)$ is an ${\mathbb R}^d$-valued Levy process, $1 \le d \le n$. We show that strong existence and uniqueness together with $L^p$-Lipschitz dependence on the initial condition $x $ imply Davie's uniqueness or path by path uniqueness. This extends a result of [E. Priola, AIHP, 2018] proved when $n=d$, $A=0$ and $\sigma(t) \equiv I $. We apply the result to some singular degenerate SDEs associated to the kinetic transport operator $ \frac{1}{2} \triangle_v f + $ ${v \cdot \partial_{x}f} $ $+F(x,v)\cdot \partial_{v}f $ when $n =2d $ and $L$ is an ${\mathbb R}^d$-valued Wiener process. For such equations strong existence and uniqueness are known under Holder type conditions on $b$. We show that in addition also Davie's uniqueness holds.

In order to achieve more accurate predicted RUL in the early stage of degradation, a novel remaining useful life (RUL) prediction method for the stochastic degradation process is proposed. Technically, modeling the degradation process as a Wiener process (WP) whose drift increment is a weighted sum of kernel functions can flexibly depict the nonlinear degradation trend. Introducing a long short term memory (LSTM) network can capture the long-term dependencies of the offline experimental and online observed degradation data to forecast the future degradation increment. Then, based on the degradation model, a numerical approximate distribution of the RUL is derived to quantify the uncertainty of the predicted RUL. Finally, a practical case study of lithium-ion batteries is provided to demonstrate the high accuracy of the proposed methods for RUL prediction especially in the early stage of degradation.

The parallel constant-stress accelerated degradation test (PCSADT) is a popular method used to assess the reliability of highly reliable products in a timely manner. Although the maximum likelihood (ML) method is commonly utilized to estimate the PCSADT parameters, the explicit forms of the ML estimators, and their corresponding Fisher information matrix are usually difficult to obtain. In this article, we propose a two-stage ML (TSML) estimation procedure for a time-transformed model. In the proposed procedure, all the TSML estimators not only have explicit expressions but also possess consistency and asymptotic normality. Hence, this method is tractable for reliability engineers. Furthermore, the TSML estimators can provide constructive information about the unknown accelerated relationship law. The proposed method is also applied to analyze light-emitting diode data and compare the performance of our estimation procedures with the ML method via simulations.

Let X X be the solution of a stochastic differential equation in Euclidean space driven by standard Brownian motion, with measurable drift and Sobolev diffusion coefficient. In our main result we show that when the drift is measurable and the diffusion coefficient belongs to an appropriate Sobolev space, the law of X X satisfies Talagrand’s inequality with respect to the uniform distance.

In this article, a stochastic Leontief model with impulses has been studied, which is represented by a system of stochastic differential-algebraic equations, in both sides of the rectangular consta...

This work deals with LQR optimal control problems where the randomness of the dynamic systems evolves according to: (a) multiplicative and additive white noise, (b) mixture white noise and (c) white noise and colored noise. Total cost quadratic and discounted cost quadratic are studied. The white noise is represented by the Wiener process whereas the colored noise is represented by the solution of the Ornstein-Uhlenbeck’s stochastic dierential equation, whit initial condition is constant or normally distributed. In order to find the value functions and optimal policies, as well as algebraic and stochastic Riccati equations the dynamic programming technique was used. The theoretical results are illustrated by two applications. Numerical simulations are carried out using Matlab.

This manuscript extends the literature on the application of geometric Brownian motion. Forecasted drift and diffusion terms estimated separately and recursively are plugged into the framework to forecast S&P500 index values. Expected index values are estimated from 100,000 simulated index values and probabilities. The results of comparing expected index values with actual values indicate that although reliable predictions of S&P500 index values can be obtained at monthly, quarterly, and annual frequencies, the reliability may decrease in that order.

We find new conditions for existence of strong solutions of ordinary differential equations with random right-hand side, stochastic differential equations with measurable random drift, and their trajectory analogs with symmetric integrals. We show that solutions of Ito equations satisfy a parabolic equation along trajectories of a Wiener process.

Understanding the distribution of an event duration time is essential in many studies. The exact time to the event is often unavailable, and thus so is the full event duration. By linking relevant longitudinal measures to the event duration, we propose to estimate the duration distribution via the first-hitting-time model (e.g. Lee and Whitmore in Stat Sci 21(4):501–513, 2006). The longitudinal measures are assumed to follow a Wiener process with random drift. We apply a variant of the MCEM algorithm to compute likelihood-based estimators of the parameters in the longitudinal process model. This allows us to adapt the well-known empirical distribution function to estimate the duration distribution in the presence of missing time origin. Estimators with smooth realizations can then be obtained by conventional smoothing techniques. We establish the consistency and weak convergence of the proposed distribution estimator and present its variance estimation. We use a collection of wildland fire records from Alberta, Canada to motivate and illustrate the proposed approach. The finite-sample performance of the proposed estimator is examined by simulation. Viewing the available data as interval-censored times, we show that the proposed estimator can be more efficient than the well-established Turnbull estimator, an alternative that is often applied in such situations. Supplementary Information The online version contains supplementary material available at 10.1007/s10985-021-09520-w.

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

With the aim of solving the reliability modeling and calculation of multivariate stochastic degradation systems, two stochastic degradation models based on the bivariate Wiener process are proposed, in which two performance characteristics are composited to one variable. Two different failure modes including the defect-based failure and the duration-based failure are considered. The explicit expressions of the system reliability are derived in the cases that the performance characteristics are not composited or the performance characteristics are composited according to the linear combination of the degradation measurements. An algorithm based on the Monte Carlo simulation is proposed to simulate the degradation process, in which the performance characteristics are composited in arbitrary forms, and the correctness of the analytical results is also verified. Finally, some numerical examples are presented to illustrate the present reliability assessment method。

In physics, chemistry, and mathematics, the process of Brownian motion is often identified with the Wiener process that has infinitesimal increments. Recently, many models of Brownian motion with finite velocity have been intensively studied. We consider one of such models, namely, a generalization of the Goldstein–Kac process to the three-dimensional case with the Erlang-2 and Maxwell–Boltzmann distributions of velocities alternations. Despite the importance of having a three-dimensional isotropic random model for the motion of Brownian particles, numerous research efforts did not lead to an expression for the probability of the distribution of the particle position, the motion of which is described by the threedimensional telegraph process. The case where a particle carries out its movement along the directions determined by the vertices of a regular n + 1-hedron in the n-dimensional space was studied in [13], and closed-form results for the distribution of the particle position were obtained. Here, we obtain expressions for the distribution function of the norm of the vector that defines particle’s position at renewal instants in semi-Markov cases of the Erlang-2 and Maxwell–Boltzmann distributions and study its properties. By knowing this distribution, we can determine the distribution of particle positions, since the motion of a particle is isotropic, i.e., the direction of its movement is uniformly distributed on the unit sphere in ℝ3. Our results may be useful in studying the properties of an ideal gas.

We study the focusing stochastic nonlinear Schrodinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the $$L^2$$ -critical and supercritical cases. The mass ( $$L^2$$ -norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate both theoretically and numerically how the energy is affected by various spatially correlated random perturbations and its dependence on the discretization parameters and the schemes. We then perform numerical investigation of the noise influence on the global dynamics measuring the probability of blow-up versus scattering behavior depending on parameters of correlation kernels. Finally, we study numerically the effect of the spatially correlated noise on the blow-up behavior, and conclude that such random perturbations do not influence the blow-up dynamics, except for shifting of the blow-up center location. This is similar to what we observed for a space-time white driving noise in Millet et al. (Numerical study of solutions behavior to the 1d stochastic $$L^2$$ -critical and supercritical nonlinear Schrodinger equation, 2020. arXiv:2006.10695 ).

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.

Degradation tests have been used for the purpose of assessing reliability information. In this paper, we consider a degradation test design problem under a statistical model mis-specification scena...

Let {X(t),t≥0} be a Wiener process with infinitesimal mean μ and variance 1, starting at X(0)∈[0,1]. Assume that X(0) is a random variable and define τ to be the first time that X(t) = 0 or 1. We l...