模糊二阶矩过程及均方收敛性

基于可信性测度,定义了新的模糊二阶矩过程,证明了该二阶矩过程存在方差函数和协方差函数.对于在可信性空间(Θ,()(Θ),Cr)上具有二阶矩的模糊变量集合H,证明了其对通常的线性运算封闭,并在H中研究了模糊均方收敛的必要条件和充分条件.最后讨论了模糊均方收敛的若干性质.     

若干基于模糊随机样本的SLT定理

基于模糊随机样本空间,提出了模糊随机期望风险泛函,模糊随机经验风险泛函和模糊随机经验风险泛函最小化原则等概念;基于模糊随机变量及其期望,讨论了相关概率不等式;最后证明了基于模糊随机样本统计学习理论的关键定理并研究了学习过程一致收敛速度的界等问题.     

模糊拓扑空间的m—收敛理论

本文在模糊拓扑空间中引进了模糊网和模糊滤子的ｍ－收敛概念，探讨了模糊网的ｍ－收敛与模糊滤子的ｍ－收敛之间的关系，得到了Ｈａｕｓｄｏｒｆｆ拓扑与模糊网的ｍ－收敛和模糊滤子的ｍ－收敛之间的关系以及刻画Ｈａｕｓｄｏｒｆｆ拓扑几何特征的定理。     

模糊化的Egoroff定理

在一般模糊测度空间上,针对可测模糊值函数序列给出了几乎处处收敛,几乎一致收敛和伪几乎一致收敛的概念,并在此基础上,进一步研究了这几种收敛的蕴涵关系,从而获得了模糊化的Egoroff定理,使模糊值函数序列的理论得到进一步丰富.     

直觉模糊矩阵幂序列的收敛准则

首先,给出直觉模糊矩阵幂的隶属度和非隶属度的表达形式.其次,定义直觉模糊矩阵的最小正周期数和收敛指数,得到单增的直觉模糊矩阵幂序列是收敛的结论.最后,给出直觉模糊矩阵幂序列收敛的两个充分条件.     

Choquet积分的收敛定理

Murofushi、Sugeno等学者已对关于模糊测度的Choquet积分进行了详细的研究,但关于积分的收敛理论是不够的。本文讨论这一问题,给出Choquet积分的一些收敛定理,包括广义单调收敛定理、Fatou引理等。从中可以看出,Choquet积分与Sugeno模糊积分具有相同的收敛定理。     

i.i.d情况下非参数回归模型的误差密度估计的收敛性质

本文研究了i．i．d情况下非参数回归的误差密度估计的一致收敛和均方收敛，给出了一定条件下误差密度的估计量f^n(x)的一致收敛速度和均方收敛速度。     

模糊值Choquet积分(Ⅱ)--函数关于模糊值模糊测度的Choquet积分

研究一种取值于模糊数集的Choquet积分，该积分的被积函数是单值函数，所用的测度是模糊值模糊测度。给出其定义、性质和收敛定理。     

Fuzzy-Val模糊值Choquet积分(Ⅱ)——函数关于模糊值模糊测度的Choquet积分(英文)

研究一种取值于模糊数集的 Choquet积分 ,该积分的被积函数是单值函数 ,所用的测度是模糊值模糊测度。给出其定义、性质和收敛定理     

模糊数值函数Henstock积分的收敛定理

给出模糊数值函数Henstock积分的收敛定理,特别给出了Kaleva积分的收敛定理,该结果推广了Kaleva积分以前若干个收敛定理。     

Numerical solution of stochastic differential equations by second order Runge–Kutta methods

In this paper we propose the numerical solutions of stochastic initial value problems via random Runge–Kutta methods of the second order and mean square convergence of these methods is proved. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced. Expectation and variance of the approximating process are computed. Numerical examples show that the approximate solutions have a good degree of accuracy.     

Weak first- or second-order implicit Runge–Kutta methods for stochastic differential equations with a scalar Wiener process

New fully implicit stochastic Runge–Kutta schemes of weak order 1 or 2 are proposed for stochastic differential equations with sufficiently smooth drift and diffusion coefficients and a scalar Wiener process, which are derivative-free and which are A-stable in mean square for a linear test equation in some general settings. They are sought in a transparent way and their convergence order and stability properties are confirmed in numerical experiments.     

Equations for second moments of solutions of a system of linear differential equations with random semi-Markov coefficients and random input

We derive equations that determine second moments of a random solution of a system of Itô linear differential equations with coefficients depending on a finite-valued random semi-Markov process. We obtain necessary and sufficient conditions for the asymptotic stability of solutions in the mean square with the use of moment equations and Lyapunov stochastic functions.

     

On Estimating the Cumulant Generating Function of Linear Processes

We compare two estimates of the cumulant generating function of a stationary linear process. The first estimate is based on the empirical moment generating function. The second estimate uses the linear representation of the process and the empirical moment generating function of the innovations. Asymptotic expressions for the mean square errors are derived under short- and long-range dependence. For long-memory processes, the estimate based on the linear representation turns out to have a better rate of convergence. Thus, exploiting the linear structure of the process leads to an infinite gain in asymptotic efficiency.     

线性随机变时滞微分方程指数Euler方法的收敛性和稳定性

文应用指数Euler方法研究了线性随机变时滞微分方程的收敛性和稳定性;首先,证明了指数Euler方法是$\frac{1}{2}$阶均方收敛的;其次,在解析解均方稳定的前提下,通过跟Euler-Maruyama方法比较发现指数Euler方法在大步长下依然保持解析解的均方稳定性;最后,用数值试验验证了收敛和稳定的结果.     

Almost sure,L_1-and L_2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration

Under a first order moment condition on the immigration mechanism, we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process) converges almost surely. If an x log(x) moment condition on the branching mechanism does not hold, then the limit is zero. If this x log(x) moment condition holds, then we prove L_1 convergence as well. The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If, in addition, a suitable extra power moment condition on the branching mechanism holds, then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L_1 limit. Moreover, under a second order moment condition on the branching and immigration mechanisms, we prove L_2 convergence of an appropriately scaled process and the above-mentioned projections as well. A representation of the limits is also provided under the same moment conditions.     

随机比例方程带线性插值的半隐式Euler方法的均方收敛性

本文研究非线性随机比例方程带线性捅值的半隐式Euler方法的均方收敛性,证明了这类方法是1/2阶均方收敛的.数值试验验证了所获理论结果的正确性.     

A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation

In this paper a new version of the chain rule for calculating the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the corresponding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.     

On the order conditions of fuzzy convergence classes

By means of the order structure of the related lattice, the LIMINF condition of fuzzy convergence classes is proposed in this paper, which reflects the essential difference between fuzzy convergence classes and ordinary convergence classes. The relationship between the LIMINF condition and two related conditions proposed by Liu and Wang respectively are discussed. The theory of fuzzy convergence classes based on LIMINF condition is established for topological molecular lattices, L-topological spaces (in the sense of Chang or Lowen), weakly induced spaces, and induced spaces.     

非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性

本文研究了数值求解非自治随机微分方程的正则Euler-Maruyama分裂（CEMS）方法,该方程的漂移项系数带有刚性且允许超线性增长,扩散项系数满足全局Lipschitz条件.首先,证明了CEMS方法的强收敛性及收敛速度.其次,证明了在适当条件下CEMS方法是均方稳定的.进一步,利用离散半鞅收敛定理,研究了CEMS方法的几乎必然指数稳定性.结果表明,CEMS方法在漂移系数的刚性部分满足单边Lipschitz条件下可保持几乎必然指数稳定性.最后通过数值实验,检验了CEMS方法的有效性并证实了我们的理论结果.