THE SOLUTION OF RANDOM EIGENVALUE PROBLEM WITH SMALL STOCHASTIC PROCESSES

This paper considers an eigenvalue problem containing small stochastic processes. For every fixed is, we can use the Prufer substitution to prove the existence of the random solutions lambda(n) and u(n) in the meaning of large probability. These solutions can be expanded in epsilon regularly, and their correction terms can be obtained by solving some random linear differential equations.

Study of Volatility Stochastic Processes in the Context of Solvency Forecasting for Sri Lankan Life Insurers

The main business of Life Insurers is Long Term contractual obligations with a typical lifetime of 20 - 40 years. Therefore, the Solvency metric is defined by the adequacy of capital to service the cash flow requirements arising from the said obligations. The main component inducing volatility in Capital is market sensitive Assets, such as Bonds and Equity. Bond and Equity prices in Sri Lanka are highly sensitive to macro-economic elements such as investor sentiment, political stability, policy environment, economic growth, fiscal stimulus, utility environment and in the case of Equity, societal sentiment on certain companies and industries. Therefore, if an entity is to accurately forecast the impact on solvency through asset valuation, the impact of macro-economic variables on asset pricing must be modelled mathematically. This paper explores mathematical, actuarial and statistical concepts such as Brownian motion, Markov Processes, Derivation and Integration as well as Probability theorems such as the Probability Density Function in determining the optimum mathematical model which depicts the accurate relationship between macro-economic variables and asset pricing.

Understanding Molecular Dynamics with Stochastic Processes via Real or Virtual Dynamics

Molecular dynamics with the stochastic process provides a convenient way to compute structural and thermodynamic properties of chemical, biological, and materials systems. It is demonstrated that the virtual dynamics case that we proposed for the Langevin equation [J. Chem. Phys. 147, 184104 （2017）] in principle exists in other types of stochastic thermostats as well. The recommended ＂middle＂ scheme [J. Chem. Phys. 147, 034109 （2017）] of the Andersen thermostat is investigated as an example. As shown by both analytic and numerical results, while the real and virtual dynamics cases approach the same plateau of the characteristic correlation time in the high collision frequency limit, the accuracy and efficiency of sampling are relatively insensitive to the value of the collision frequency in a broad range. After we compare the behaviors of the Andersen thermostat to those of Langevin dynamics, a heuristic schematic representation thermostatting processes with molecular is proposed for understanding efficient stochastic dynamics.

Importance Sampling Strategy for Oscillatory Stochastic Processes

This paper contributes to the structural reliability problem by presenting a novel approach that enables for identification of stochastic oscillatory processes as a critical input for given mechanical models. Identification development follows a transparent image processing paradigm completely independent of state-of-the-art structural dynamics, aiming at delivering a simple and wide purpose method. Validation of the proposed importance sampling strategy is based on multi-scale clusters of realizations of digitally generated non-stationary stochastic processes. Good agreement with the reference pure Monte Carlo results indicates a significant potential in reducing the computational task of first passage probabilities estimation, an important feature in the field of e.g., probabilistic seismic design or risk assessment generally.

Applications of Dynamic-Equilibrium Continuous Markov Stochastic Processes to Elements of Survival Analysis

In this article, we summarize some results on invariant non-homogeneous and dynamic-equilibrium (DE) continuous Markov stochastic processes. Moreover, we discuss a few examples and consider a new application of DE processes to elements of survival analysis. These elements concern the stochastic quadratic-hazard-rate model, for which our work 1) generalizes the reading of its It? stochastic ordinary differential equation (ISODE) for the hazard-rate-driving independent (HRDI) variables, 2) specifies key properties of the hazard-rate function, and in particular, reveals that the baseline value of the HRDI variables is the expectation of the DE solution of the ISODE, 3) suggests practical settings for obtaining multi-dimensional probability densities necessary for consistent and systematic reconstruction of missing data by Gibbs sampling and 4) further develops the corresponding line of modeling. The resulting advantages are emphasized in connection with the framework of clinical trials of chronic obstructive pulmonary disease (COPD) where we propose the use of an endpoint reflecting the narrowing of airways. This endpoint is based on a fairly compact geometric model that quantifies the course of the obstruction, shows how it is associated with the hazard rate, and clarifies why it is life-threatening. The work also suggests a few directions for future research.

Stochastic processes shape the functional and phylogenetic structure of bird assemblages at the mine area in southwest China

Understanding the mechanisms of community assembly is a key question in ecology.Metal pollution may result in significant changes in bird community structure and diversity,with implications for ecosystem processes and function.However,the relative importance of these pro-cesses in shaping the bird community at the polluted area is still not clear.Here,we explored bird species richness,functional,and phylogenetic diversity,and the assembly processes of community at the mine region of southwest China.Our results showed that the 3 dimensions of diversity at the mine area were lower than that at the reference sites.In the community assembly,the result was O<NRI/NFR1<1.96,which indicated deterministic processes(environmental filtering)might drive community clustering.The results of the neutral community model,and normalized stochasticity ratio,showed the dominant role of stochastic processes in shaping the bird community assembly.We further quanti-fied the community-level habitat niche breadth(Bcom),and we found that there was no difference in Bcom-value between the mine area and reference sites.This indicates that the bird communities at the mine area and 3 reference sites were not subjected to extreme environmental selection(same or different resource allocation)to form a highly specialized niche.These findings provide insights into the distribution patterns and dominant ecological processes of bird communities under metal exposure,and extend the knowledge in community assembly mechanisms of bird communities living in the mine area.

Nitrogen deposition mediates more stochastic processes in structuring plant community than soil microbial community in the Eurasian steppe

Anthropogenic environmental changes may affect community assembly through mediating both deterministic(e.g.,competitive exclusion and environmental filtering)and stochastic processes(e.g.,birth/death and dispersal/colonization).It is traditionally thought that environmental changes have a larger mediation effect on stochastic processes in structuring soil microbial community than aboveground plant community;however,this hypothesis remains largely untested.Here we report an unexpected pattern that nitrogen(N)deposition has a larger mediation effect on stochastic processes in structuring plant community than soil microbial community(those<2 mm in diameter,including archaea,bacteria,fungi,and protists)in the Eurasian steppe.We performed a ten-year nitrogen deposition experiment in a semiarid grassland ecosystem in Inner Mongolia,manipulating nine rates(0–50 g N m^(-2)per year)at two frequencies(nitrogen added twice or 12 times per year)under two grassland management strategies(fencing or mowing).We separated the compositional variation of plant and soil microbial communities caused by each treatment into the deterministic and stochastic components with a recently-developed method.As nitrogen addition rate increased,the relative importance of stochastic component of plant community first increased and then decreased,while that of soil microbial community first decreased and then increased.On the whole,the relative importance of stochastic component was significantly larger in plant community(0.552±0.035;mean±standard error)than in microbial community(0.427±0.035).Consistently,the proportion of compositional variation explained by the deterministic soil and community indices was smaller for plant community(0.172–0.186)than microbial community(0.240–0.767).Meanwhile,as nitrogen addition rate increased,the linkage between plant and microbial community composition first became weaker and then became stronger.The larger stochasticity in plant community relative to microbial community assembly suggested that more stochastic strategies(e.g.,seeds addition)should be adopted to maintain above-than below-ground biodiversity under the pressure of nitrogen deposition.

Approximation of weak sense stationary stochastic processes from local averages

We show that a weak sense stationary stochastic process can be approximated by local averages. Explicit error bounds are given. Our result improves an early one from Splettst?sser.

Stochastic processes in renewable power systems: From frequency domain to time domain

With the increasing penetration of renewable energy resources(RESs), the uncertainties of volatile renewable generations significantly affect the power system operation. Such uncertainties are usually modeled as stochastic variables obeying specific distributions by neglecting the temporal correlations. Conventional approaches to hedge the negative effects caused by such uncertainties are thus hard to pursue a trade-off between computation efficiency and optimality. As an alternative, the theory of stochastic process can naturally model temporal correlation in closed forms. Attracted by this feature, our research group has been conducting thorough researches in the past decade to introduce stochastic processes within renewable power systems. This paper summarizes our works from the perspective of both the frequency domain and the time domain, provides the tools for the analysis and control of power systems under a unified framework of stochastic processes, and discusses the underlying reasons that stochastic process-based approaches can perform better than conventional approaches on both computational efficiency and optimality. These work may shed a new light on the research of analysis, control and operation of renewable power systems.Finally, this paper outlooks the theoretic developments of stochastic processes in future’s renewable power systems.

Representation Theorems for Fuzzy Random Sets and Fuzzy Stochastic Processes

The fuzzy static and dynamic random phenomena in an abstract separable Banach space is discussed in this paper. The representation theorems for fuzzy set valued random sets, fuzzy random elements and fuzzy set valued stochastic processes are obtained.

Reliability Analysis of Repairable Systems Using Stochastic Point Processes

In order to analyze the failure data from repairable systems, the homogeneous Poisson process (HPP) is usually used. In general, HPP cannot be applied to analyze the entire life cycle of a complex, re-pairable system because the rate of occurrence of failures (ROCOF) of the system changes over time rather than remains stable. However, from a practical point of view, it is always preferred to apply the simplest method to address problems and to obtain useful practical results. Therefore, we attempted to use the HPP model to analyze the failure data from real repairable systems. A graphic method and the Laplace test were also used in the analysis. Results of numerical applications show that the HPP model may be a useful tool for the entire life cycle of repairable systems.

Robust H-infinity control of uncertain stochastic time-delay linear repetitive processes

Repetitive processes are a distinct class of 2D systems of both theoretic and practical interest.The robust H-infinity control problem for uncertain stochastic time-delay linear continuous repetitive processes is investigated in this paper.First,sufficient conditions are proposed in terms of stochastic Lyapunov stability theory,It o differential rule and linear matrix inequality technology.The corresponding controller design is then cast into a convex optimization problem.Attention is focused on constructing an admissible controller,which guarantees that the closed-loop repetitive processes are mean-square asymptotically stable and have a prespecified H-infinity performance γ with respect to all energy-bounded input signals.A numerical example illustrates the effectiveness of the proposed design scheme.

AWeighted Average Finite Difference Scheme for the Numerical Solution of Stochastic Parabolic Partial Differential Equations

In the present paper,the numerical solution of It?type stochastic parabolic equation with a timewhite noise process is imparted based on a stochastic finite difference scheme.At the beginning,an implicit stochastic finite difference scheme is presented for this equation.Some mathematical analyses of the scheme are then discussed.Lastly,to ascertain the efficacy and accuracy of the suggested technique,the numerical results are discussed and compared with the exact solution.

Dynamics of Advantageous Mutant Spread in Spatial Death-Birth and Birth-Death Moran Models

The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the Moran model, individuals first reproduce according to their fitness and then replace a neighboring individual. From a biological standpoint, the opposite dynamics, where individuals first die and are then replaced by a neighboring individual according to its fitness, are equally relevant. Here, we investigate this death-birth analogue of the biased voter model. We construct the process mathematically, derive the associated dual process, establish bounds on the survival probability of a single mutant, and prove that the process has an asymptotic shape. We also briefly discuss alternative birth-death and death-birth dynamics, depending on how the mutant fitness advantage affects the dynamics. We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model, whereas the birth-death model is fundamentally different from the death-birth model when fitness affects the latter event.

Stochastic Responses of a Shell of Rotational Symmetry Haing Physically Nonlinear Behavior

Concrete is considered as an elastic body when subjected to compressive stress butas a brittle body when subjected to tensile stress. That is, after the elastic extremum it cannotcontinue to bear the stress. Based on this.'cracked' physical model, for stochastic extemalwind loads, nonlinear stochastic constitutive equations have been established. To solve equations of nonlinear stochastic vibration, an iterative statistic linearization method (ISLM) hasbeen developedb. Some statistic characteristics were foundd out.

Numerical Simulations for Stochastic Computer Virus Propagation Model

We are presenting the numerical simulations for the stochastic computer virus propagation model in this manuscript.We are comparing the solutions of stochastic and deterministic computer virus models.Outcomes of a threshold number R0 hold in stochastic computer virus model.If R_(0)<1 then in such a condition virus controlled in the computer population while R_(0)>1 shows virus rapidly spread in the computer population.Unfortunately,stochastic numerical techniques fail to cope with large step sizes of time.The suggested structure of the stochastic non-standard finite difference technique can never violate the dynamical properties.On this basis,we can suggest a collection of strategies for removing virus’s propagation in the computer population.

Path-dependent speciation in dynamic fitness landscapes

Species is a fundamental concept in evolutionary biology and biodiversity.However,existing species definitions are often influenced by artificial factors or are challenging in practical application,leading to confusion in species classification.Due to uncertain environmental changes and random genetic drift,the fitness expectations of a population may shift,causing species to evolve to a new evolutionary state based on their current instantaneous fitness within a dynamic fitness landscape.This contrasts with the classic static fitness landscape,where fitness expectations are constant.In a dynamic fitness landscape,speciation may exhibit path dependence,where the evolution of traits follows a probabilistic path,creating feedback that shapes evolutionary trajectories.The path-dependent evolutionary mechanism suggests that species survival within an ecosystem is not directly determined by their fitness but by the probability of their evolutionary pathways.This model also indicates that species can coexist with varying probabilities under limited environmental pressures.Consequently,new species,cryptic species,or sympatric species may emerge via path-dependent evolutionary processes.Within this framework,we developed a mathematical species concept,which may guide future species classification methodologies.

Stochastic systems with cross-correlated Gaussian white noises

This paper theoretically investigates three stochastic systems with cross-correlation Gaussian white noises. Both steady state properties of the stochastic nonlinear systems and the nonequflibrium transitions induced by the cross- correlated noises are studied. The stationary solutions of the Fokker-Planck equation for three specific examples are analysed. It is shown explicitly that the cross-correlation of white noises can induce nonequilibrium transitions.

A Stochastic Optimal Control Theory to Model Spontaneous Breathing

Respiratory variables, including tidal volume and respiratory rate, display significant variability. The probability density function (PDF) of respiratory variables has been shown to contain clinical information and can predict the risk for exacerbation in asthma. However, it is uncertain why this PDF plays a major role in predicting the dynamic conditions of the respiratory system. This paper introduces a stochastic optimal control model for noisy spontaneous breathing, and obtains a Shr&ouml;dinger’s wave equation as the motion equation that can produce a PDF as a solution. Based on the lobules-bronchial tree model of the lung system, the tidal volume variable was expressed by a polar coordinate, by use of which the Shr&ouml;dinger’s wave equation of inter-breath intervals (IBIs) was obtained. Through the wave equation of IBIs, the respiratory rhythm generator was characterized by the potential function including the PDF and the parameter concerning the topographical distribution of regional pulmonary ventilations. The stochastic model in this study was assumed to have a common variance parameter in the state variables, which would originate from the variability in metabolic energy at the cell level. As a conclusion, the PDF of IBIs would become a marker of neuroplasticity in the respiratory rhythm generator through Shr?dinger’s wave equation for IBIs.

Stochastic stability of the derivative unscented Kalman filter

This is the second of two consecutive papers focusing on the filtering algorithm for a nonlinear stochastic discretetime system with linear system state equation. The first paper established a derivative unscented Kalman filter（DUKF） to eliminate the redundant computational load of the unscented Kalman filter（UKF） due to the use of unscented transformation（UT） in the prediction process. The present paper studies the error behavior of the DUKF using the boundedness property of stochastic processes. It is proved that the estimation error of the DUKF remains bounded if the system satisfies certain conditions. Furthermore, it is shown that the design of the measurement noise covariance matrix plays an important role in improvement of the algorithm stability. The DUKF can be significantly stabilized by adding small quantities to the measurement noise covariance matrix in the presence of large initial error. Simulation results demonstrate the effectiveness of the proposed technique.