Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.     

实值非负函数关于集值序增函数的集值Riemann-Stieltjes积分

本文首先建立了实值非负函数关于集值序增函数的集值Riemann-Stieltjes积分,并讨论了集值Riemann-Stieltjes积分的性质,给出了集值Riemann-Stieltjes可积的充要条件,最后引入了集值Riemann-Stieltjes随机积分．     

有界可料过程关于集值平方可积鞅的集值随机积分

本文定义了一类有界可料过程关于集值平方可积鞅的集值随机积分,并研究了集植随机积分的性质。此为建立集值随机分析的理论奠定了基础。     

Set-valued processes induced by parameterised stochastic differential equations with an application to Tuned Mass Dampers

In this paper ordinary stochastic differential equations whose coefficients depend on uncertain parameters are considered. An approach is presented how to combine both types of uncertainty (stochastic excitation and parameter uncertainty) leading to set-valued stochastic processes. The latter serve as a robust representation of solutions of the underlying stochastic differential equations. The mathematical concept is applied to a problem from earthquake engineering, where it is shown how the efficiency of Tuned Mass Dampers can be realistically assessed in the presence of uncertainty. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays

In this paper, we formulate and investigate a class of memristor-based BAM neural networks with time-varying delays. Under the framework of Filippov solutions, the viability and dissipativity of solutions for functional differential inclusions and memristive BAM neural networks can be guaranteed by the matrix measure approach and generalized Halanay inequalities. Then, a new method involving the application of set-valued version of Krasnoselskii’ fixed point theorem in a cone is successfully employed to derive the existence of the positive periodic solution. The dynamic analysis in this paper utilizes the theory of set-valued maps and functional differential equations with discontinuous right-hand sides of Filippov type. The obtained results extend and improve some previous works on conventional BAM neural networks. Finally, numerical examples are given to demonstrate the theoretical results via computer simulations.     

在Pbkc（c[0,1]）与Pbkc（Lp[0,1]）取值的集值随机变量（1）

对拟连续测度空间(G,β,u)的一致有界等度连续函数族,通过包含关系,取凸包和闭包,构造了在Pbkc(c[0,1])与Pbkc(Lp[0,1])取值的集值随机变量及连续的集值映射,深化了集值随机过程理论研究.     

Stochastic morphological evolution equations

The inadequacy of locally defined set-valued differential equations to describe the evolution of shapes and morphological forms in biology, which are usually neither convex or nondecreasing, was recognised by J.-P. Aubin, who introduced morphological evolution equations, which are essentially nonlocally defined set-valued differential equations with the inclusion vector field also depending on the entire reachable set. This concept is extended here to the stochastic setting of set-valued Itô evolution equations in Hilbert spaces. Due to the nonanticipative nature of Itô calculus, the evolving reachable sets are nonanticipative nonempty closed random sets. The existence of solutions and their dependence on initial data are established. The latter requires the introduction of a time-oriented semi-metric in time-space variables. As a consequence the stochastic morphological evolution equations generate a deterministic nonautonomous dynamical system formulated as a two-parameter semigroup with the complication that the random subsets take values in different spaces at different time instances due to the nonanticipativity requirement. It is also shown how nucleation processes can be handled in this conceptual framework.     

Measurable solutions for stochastic evolution equations without uniqueness

A method for obtaining measurable solutions to stochastic evolution equations in which there is no uniqueness for the corresponding non-stochastic equation is presented. It involves a technique based on a measurable selection theorem for set-valued functions. No assumptions are needed on the underlying probability space. An application is given to the stochastic Navier–Stokes problem in arbitrary dimensions. We also show the existence of measurable solutions to stochastic ordinary differential equations in which there is no uniqueness. A finite-dimensional generalization is given to adapted solutions in the case of a normal filtration and path uniqueness.     

Stochastic recursive inclusions with non-additive iterate-dependent Markov noise

In this paper we study the asymptotic behaviour of stochastic approximation schemes with set-valued drift function and non-additive iterate-dependent Markov noise. We show that a linearly interpolated trajectory of such a recursion is an asymptotic pseudotrajectory for the flow of a limiting differential inclusion obtained by averaging the set-valued drift function of the recursion w.r.t. the stationary distributions of the Markov noise. The limit set theorem by Benaim is then used to characterize the limit sets of the recursion in terms of the dynamics of the limiting differential inclusion. We then state two variants of the Markov noise assumption under which the analysis of the recursion is similar to the one presented in this paper. Scenarios where our recursion naturally appears are presented as applications. These include controlled stochastic approximation, subgradient descent, approximate drift problem and analysis of discontinuous dynamics all in the presence of non-additive iterate-dependent Markov noise.     

φ-单调型集值变分包含解的Ishikawa迭代逼近

在实自反的Banach空间中引入和研究了一类新的非线性φ-单调型集值变分包含,并证明了此类变分包含解的唯一性及其具误差项的Ishikawa迭代程序的收敛性.本文结果在多方面改进和推广了Huang和Fang等人的一些相关结果.     

集值Lebesgue—Stieltjes积分

本文首先刻划了B(R_ )上的集值测度,其次建立了(R_ B(R_ ))上的集值Lebesgue-Stieltjes积分.最后,进—步建立了集值随机Lebesgue-Stietjes积分的理论.     

On set-valued stochastic integrals in an M-type 2 Banach space

In a separable Banach space, for set-valued martingale, several equivalent conditions based on the measurable selections are discussed, and then, in an M-type 2 Banach space, at first we define single valued stochastic integral by the differential of a real valued Brownian motion, after that extend it to set-valued case. We prove that the set-valued stochastic integral becomes a set-valued submartingale, which is different from single valued case, and obtain the Castaing representation theorem for the set-valued stochastic integral, which is applicable for set-valued stochastic differential equations.     

On asymptotic stability of solutions of stochastic differential equations with constant drift

In this paper we provide conditions for asymptotic stability of solutions of stochastic differential equations with constant drift. The main result of the paper extends the results of Khasminskii and Nevelson ([5]), where only the stochastic differential equations with recurrent solutions were studied     

Khasminskii-type theorems for stochastic functional differential equations with infinite delay

The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for stochastic functional equations with infinite delay. The main aim of this paper is to establish the existence-and-uniqueness theorems of global solutions for stochastic functional differential equations with infinite delay.     

Existence and Uniqueness of Solutions to Time-delays Stochastic Fractional Differential Equations with Non-Lipschitz Coefficients

In this paper, we consider the existence and uniqueness of solutions to time-varying delays stochastic fractional differential equations (SFDEs) with non-Lipschitz coefficients. By using fractional calculus and stochastic analysis, we can obtain the existence result of solutions for stochastic fractional differential equations.     

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.     

Stochastic differential inclusions and diffusion processes

Connections between weak solutions of stochastic differential inclusions and solutions of partial differential inclusions, generated by given set-valued mappings are considered. The main results are based on some continuous approximation selection theorem and weak compactness of the set of all weak solutions to a given stochastic differential inclusion.     

一类具有无穷时滞中立型非稠定脉冲随机泛函微分方程积分解的存在性

本文讨论了一类具有无穷时滞中立型非稠定脉冲随机泛函微分方程,利用Sadovskii不动点原理等工具得到了其积分解的存在性,给出其在一类二阶无穷时滞中立型非稠定脉冲随机偏微分方程积分解的存在性中的应用.     

二层多目标最优化问题解集的连通性

研究了二层多目标最优化模型(BLMOP)解集的连通性问题,其中(BLMOP)的上层集值目标函数由下层问题的有效点确定.把(BLMOP)看作成单层的集值函数优化问题,借助集值函数优化问题各种有效解集的连通性的结论,得到了(BLMOP)相应的有效解集连通性的结论.     

Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations

Fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering. This paper is concerned with the square-mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a Hilbert space. The main objective of this paper is to establish the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations. A new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy. Finally, an example is provided to illustrate the obtained theory.