Stochastic stability and bifurcation for the chronic state in Marchuk’s model with noise

A stochastic differential equation modelling a Marchuk’s model is investigated. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. Firstly, the stochastic Marchuk’s model has been simplified by applying stochastic center manifold and stochastic average theory. Secondly, by using Lyapunov exponent and singular boundary theory, we analyze the local stochastic stability and global stochastic stability for stochastic Marchuk’s model, respectively. Thirdly, we explore the stochastic bifurcation of the stochastic Marchuk’s model according to invariant measure and stationary probability density. Some new criteria ensuring stochastic pitchfork bifurcation and P-bifurcation for stochastic Marchuk’s model are obtained, respectively.     

Long term behaviors of stochastic single-species growth models in a polluted environment II

This is a continuation of our paper [M. Liu, K. Wang, X. Liu. Long term behaviors of stochastic single-species growth models in a polluted environment. Appl Math Model 2011;35:752–62]. This work still devotes to studying three stochastic single-species models in a polluted environment. For the first system, sufficient criteria for extinction, stochastic non-persistence in the mean, stochastic weak persistence in the mean, stochastic strong persistence in the mean and stochastic permanence of the population are established. The threshold between stochastic weak persistence in the mean and extinction is obtained. For the second model, sufficient conditions for extinction, stochastic non-persistence in the mean, stochastic weak persistence, stochastic weak persistence in the mean, stochastic strong persistence in the mean and stochastic permanence are established. The threshold between stochastic weak persistence and extinction is derived. For the third system, the threshold between stochastic weak persistence and extinction is obtained.     

Stability for multi-linked stochastic delayed complex networks with stochastic hybrid impulses by Dupire Itô’s formula

In this paper, the mean-square exponential stability is investigated for multi-linked stochastic delayed complex networks with stochastic hybrid impulses. Distinct from the existing literature, we study the MSDCNs on the basis of the multi-linked stochastic functional differential equations that consider the impact of a certain past interval on the present. Moreover, the stochastic hybrid impulses we discuss possess stochastic impulsive moments and impulsive gain, which make the impulses fit better to the real-world demands for control. Also, a novel concept of average stochastic impulsive gain is proposed to measure the intensity of the stochastic hybrid impulses. By the use of Dupire Itô’s formula, based on Lyapunov method, graph theory and stochastic analysis techniques, two sufficient criteria for the mean-square exponential stability are derived, which are closely related to average stochastic impulsive gain, stochastic disturbance strength as well as the topological structure of the network itself. Finally, an application about neural networks is discussed and corresponding numerical example is presented to demonstrate the feasibility and effectiveness of the theoretical results.     

两类随机控制风险秩序的等价性分析

本文首先对比分析了两类风险秩序:随机控制秩序与对偶随机控制秩序.得到并证明了下述命题:(1)效用自由秩序等价于随机控制秩序;(2)畸变自由秩序等价于对偶随机控制秩序;(3)第一、第二阶随机控制秩序等价于第一、第二阶的对偶随机控制秩序,但对高于三阶的情况由实例说明不一定成立.     

Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems

A nonlinear stochastic optimal time-delay control strategy for quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal control problem of quasi-integrable Hamiltonian system with time-delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed feedback control forces are approximated by the control forces without time-delay and the original problem is converted into a stochastic optimal control problem without time-delay. After that, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. As an example, the stochastic time-delay optimal control of two coupled van der Pol oscillators under stochastic excitation is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.     

随机DEA期望值模型的一些性质

为判别决策单元在随机DEA期望值模型下的随机有效性,首次提出了随机期望无效、随机期望弱有效、随机期望有效以及随机期望超有效的概念．并给出了三个命题用于判别不同显著性水平下随机期望效率与期望效率的关系．在此基础上,得到了两个重要的性质：（1）当期望效率保持不变时,随机期望效率为显著性水平的增函数;（2）当显著性水平保持不变时,随机期望效率为期望效率的增函数．最后,利用随机模拟和一个算例对上述结论进行了验证．     

On a new set-valued stochastic integral with respect to semimartingales and its applications

We present a new approach to a concept of a set-valued stochastic integral with respect to semimartingales. Such an integral, called set-valued stochastic up-trajectory integral, is compatible with the decomposition of the semimartingale. Some properties of this integral are stated. We show applicability of the new integral in set-valued stochastic integral equations driven by multidimensional semimartingales. The uniqueness theorem is presented. Then we extend the notion of the set-valued stochastic up-trajectory integral to definition of a fuzzy stochastic up-trajectory integral with respect to semimartingales. A result on uniqueness of a solution to fuzzy stochastic integral equations incorporating the new fuzzy stochastic up-trajectory integral driven by the multidimensional semimartingale is stated.     

Global stability and stabilization of more general stochastic nonlinear systems

This paper considers the global stability and stabilization of more general stochastic nonlinear systems. Due to the absence of the conventional assumptions (e.g., Lipschitz condition), the stochastic nonlinear systems under investigation may have more than one weak solution. However, the most associated results are only applicable to the stochastic systems having a unique strong solution, and therefore, it is meaningful to refine and extend the relevant concepts and methods to the more general case. In this paper, the concepts of stochastic stability in the more general sense are first introduced to cover the stochastic nonlinear systems having more than one weak solution. Then, the generalized stochastic Barbashin–Krasovskii theorem and LaSalle theorem are established, which present the criterions of stochastic stability for more general stochastic nonlinear systems. As one of the main contributions in this paper, we rigorously prove the generalized stochastic Barbashin–Krasovskii theorem. Moreover, based on the generalized theorems, the output-feedback and state-feedback stabilization are accomplished respectively for two classes of high-order stochastic nonlinear systems under rather weaker assumptions comparing to the existing literature.     

Stochastic Approaches to Uncertainty Quantification in CFD Simulations

This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.     

Stochastic approaches to uncertainty quantification in CFD simulations

This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.     

Methods for derivation of the stochastic Boltzmann hierarchy

We consider different methods for the derivation of the stochastic Boltzmann hierarchy corresponding to the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of hard spheres. Solutions of the stochastic Boltzmann hierarchy are the Boltzmann-Grad limit of solutions of the BBGKY hierarchy of hard spheres in the entire phase space. A new concept of reduced distribution functions corresponding to the stochastic dynamics are introduced. They take into account the contribution of the hyperplanes of lower dimension where stochastic point particles interact with one another. The solutions of the Boltzmann equation coincide with one-particle distribution functions of the stochastic Boltzmann hierarchy and are represented by integrals over the hyperplanes where the stochastic point particles interact with one another.     

Numerical comparison of three stochastic methods for nonlinear PN junction problems

We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.     

Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment

This paper intends to develop a new method to obtain the threshold of an impulsive stochastic chemostat model with saturated growth rate in a polluted environment. By using the theory of impulsive differential equations and stochastic differential equations, we obtain conditions for the extinction and the permanence of the microorganisms of the deterministic chemostat model and the stochastic chemostat model. We develop a new numerical computation method for impulsive stochastic differential system to simulate and illustrate our theoretical conclusions. The biological results show that a small stochastic disturbance can cause the microorganism to die out, that is, a permanent deterministic system can go to extinction under the white noise stochastic disturbance. The theoretical method can also be used to explore the threshold of some impulsive stochastic differential equations.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coefficients via a Doss–Sussmann-type transformation, so that a stochastic viscosity solution can be defined in a “point-wise” manner. Using the recently developed theory on backward/backward doubly stochastic differential equations, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman–Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscosity solution will be addressed separately in Part II where the relation between the stochastic viscosity solution and the ω-wise, “deterministic” viscosity solution to the PDE with random coefficients will be established.     

Optimal control for two-dimensional stochastic second grade fluids

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional. We show that the Gâteaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.     

Small time and semiclassical asymptotics for stochastic heat equations driven by Lévy noise

The paper is devoted to the study of stochastic heat equations driven by Lévy noise. Applying the WKB method, we obtain multiplicative small time and semiclassical asymptotics for the Green functions and for solutions of the Cauchy problem for the heat equation under some natural additional assumptions on their coefficients. The first step in this construction consists in solving the corresponding stochastic Hamilton-Jacobi equations which constitute the "classical part" of the semiclassical approximation. In its turn, the corresponding Hamilton-Jacobi equations can be solved via solutions of the corresponding Hamiltonian systems, which gives rise to the method of stochastic characteristics. The relevant theory of stochastic Hamiltonian systems and stochastic Hamilton-Jacobi equations was developed in our previous papers. Here we put the final rung on the ladder: stochastic Hamiltonian systems, stochastic Hamilton-Jacobi equations, stochastic heat equations.     

Bogoliubov averaging principle of stochastic reaction–diffusion equation

The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation.     

On nonsmooth and discontinuous problems of stochastic systems optimization

A class of stochastic optimization problems is analyzed that cannot be solved by deterministic and standard stochastic approximation methods. We consider risk-control problems, optimization of stochastic networks and discrete event systems, screening irreversible changes, and pollution control. The results of Ermoliev et al. are extended to the case of stochastic systems and general constraints. It is shown that the concept of stochastic mollifier gradient leads to easily implementable computational procedures for systems with Lipschitz and discontinuous objective functions. New optimality conditions are formulated for designing stochastic search procedures for constrained optimization of discontinuous systems.     

Natural gas cash-out problem: Bilevel stochastic optimization approach

A stochastic formulation of the natural gas cash-out problem is given in a form of a bilevel multi-stage stochastic programming model with recourse. After reducing the original formulation to a bilevel linear problem, a stochastic scenario tree is defined by its node events, and time series forecasting is used to produce stochastic values for data of natural gas price and demand. Numerical experiments were run to compare the stochastic solution with the perfect information solution and the expected value solutions.     

Resilient and robust finite-time H∞ control for uncertain discrete-time jump nonlinear systems

This paper studies the robust and resilient finite-time H_∞ control problem for uncertain discrete-time nonlinear systems with Markovian jump parameters. With the help of linear matrix inequalities and stochastic analysis techniques, the criteria concerning stochastic finite-time boundedness and stochastic H_∞ finite-time boundedness are initially established for the nonlinear stochastic model. We then turn to stochastic finite-time controller analysis and design to guarantee that the stochastic model is stochastically H_∞ finite-time bounded by employing matrix decomposition method. Applying resilient control schemes, the resilient and robust finite-time controllers are further designed to ensure stochastic H_∞ finite-time boundedness of the derived stochastic nonlinear systems. Moreover, the results concerning stochastic finite-time stability and stochastic finite-time boundedness are addressed. All derived criteria are expressed in terms of linear matrix inequalities, which can be solved by utilizing the available convex optimal method. Finally, the validity of obtained methods is illustrated by numerical examples.