Poisson跳的随机延迟微分方程Heun方法的均方收敛性

Heun方法是一类求解随机延迟微分方程的数值方法,本文试图研究Poisson跳的随机延迟微分方程Heun方法的均方收敛性.当Poisson跳的随机延迟微分方程满足一定约束条件时,获得Heun方法求解方程所得的数值解收敛于真解,且均方收敛阶为1的理论结果2.文末数值试验的结果验证了理论结果的正确性.     

非线性变延迟随机微分方程Heun法的均方稳定性

利用线性插值的改进Heun法,研究了改进Heun法用于求解非线性变延迟随机微分方程的稳定性,得到了在噪声为乘性噪声时,Heun法用于求解非线性变延迟随机微分方程的均方稳定性的充分条件,丰富了非线性延迟随机微分方程算法理论,并用MATLAB对实际算例进行了数值模拟.     

基于LMI方法的多时滞随机神经网络的指数稳定性

研究了一类具有多个时滞的随机神经网络的均方指数稳定性问题,应用Lyapunov-Krasovskii泛函稳定理论和线性矩阵不等式(LMI)方法,建立了该系统解的指数稳定判别准则,最后通过数值举例阐述了结果的有效性.     

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.     

比较原理与一类具有时滞和马尔科夫切换的随机抛物方程组的稳定性分析（英文）北大核心CSCD

研究一类具有时滞和马尔科夫切换的随机抛物方程组的均方稳定性.通过建立比较原理,运用时滞微分不等式和随机分析技巧,获得了该系统的均方稳定、均方一致稳定、均方渐近稳定和均方指数稳定.最后,给出了主要定理的一个应用实例.     

随机变时滞模糊神经网络的均方指数稳定性

利用Lyapunov泛函和随机分析的方法,研究了一类具有变时滞随机模糊细胞神经网络的均方指数稳定性,得到了这类神经网络均方指数稳定性的充分条件.数值例子说明了得到的结果的有效性.     

随机延迟微分方程的Milstein方法的非线性均方稳定性

本文针对一般的非线性随机延迟微分方程,证明了当系统理论解满足均方稳定性条件时,则当方程的漂移和扩散项满足一定的条件时,Milstein方法也是均方稳定的.数学实验进一步验证了我们的结论.     

带Poisson跳年龄相关随机时滞种群系统的均方稳定性

本文研究了带Poisson跳年龄相关随机时滞种群系统均方稳定性的问题.在一定条件下,给出了数值解均方稳定的定义.利用补偿随机θ法讨论系统数值解的均方稳定性,给出数值解稳定的充分条件.获得了当1/2≤θ≤1时,对于任意的步长?τ/m,数值解是均方稳定的;当0≤θ1,时,如果步长?t∈(0,?t0),数值解是指数均方稳定的的结果.最后通过数值算例推广并验证了结果的有效性和正确性.     

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考察一类Markov切换时变时滞随机系统的均方指数稳定性. 利用基于Liapunov函数和线性矩阵不等式的方法, 给出了使状态反馈控制系统能克服不确定性和随机干扰, 在均方意义下达到指数稳定的充分条件. 当Markov链遍历所有模态时, 给出了一个独立于Markov链模态集的增益矩阵, 使得状态反馈控制系统均方指数稳定     

分段连续型随机微分方程指数Euler方法的稳定性

给出了线性分段连续型随机微分方程指数Euler方法的均方指数稳定性.经典的对稳定性理论分析,通常应用的是Lyapunov泛函理论,然而,应用该方程本身的特点和矩阵范数的定义给出了该方程精确解的均方稳定性.以往对于该方程应用隐式Euler方法得到对于任意步长数值解的均方稳定性,而应用显式Euler方法得到了相同的结果.最后,给出实例验证结论的有效性.     

滞后型分段连续随机微分方程的稳定性

本文研究了滞后型分段连续随机微分方程的解析稳定性和数值稳定性问题.首先,利用伊藤公式等方法获得了解析解均方稳定的条件,其次,对于包括均方稳定和T-稳定在内的Euler-Maruyama方法的数值稳定性问题,运用不等式技术和随机分析方法获得了一些新的结果,证明了在一定条件下,Euler-Maruyama方法既是均方稳定又是T-稳定的,推广了随机延迟微分方程的数值稳定性结论.     

Mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations

This paper deals with the mean-square exponential stability of stochastic theta methods for nonlinear stochastic delay integro-differential equations. It is shown that the stochastic theta methods inherit the mean-square exponential stability property of the underlying system. Moreover, the backward Euler method is mean-square exponentially stable with less restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.     

Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations

There are few results on the numerical stability of nonlinear neutral stochastic delay differential equations (NSDDEs). The aim of this paper is to establish some new results on the numerical stability for nonlinear NSDDEs. It is proved that the semi-implicit Euler method is mean-square stable under suitable condition. The theoretical result is also confirmed by a numerical experiment.     

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.     

Stability of the split-step backward Euler scheme for stochastic delay integro-differential equations with Markovian switching

In this paper, we concentrate on the numerical approximation of solutions of stochastic delay integro-differential equations with Markovian switching (SDIDEsMS). We establish the split-step backward Euler (SSBE) scheme for solving linear SDIDEsMS and discuss its convergence and stability. Moreover, the SSBE method is convergent with strong order γ = 1/2 in the mean-square sense. The conditions under which the SSBE method is mean-square stable and general mean-square stable are obtained. Some illustrative numerical examples are presented to demonstrate the stability of the numerical method and show that SSBE method is superior to Euler method.     

Asymptotic mean-square stability of linear multi-step methods for SODEs

In this article we present results of a linear stability analysis of stochastic linear multi-step methods for stochastic ordinary differential equations. As in deterministic numerical analysis we use a linear time-invariant test equation and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution of that test equation. Sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods are obtained with the aide of Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams-Bashforth- and Adams-Moulton-methods and the BDF method. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

线性中立型随机延迟微分方程Euler方法的均方稳定性

本文讨论Euler方法用于求解线性中立型随机延迟微分方程初值问题时数值解的稳定性,利用了一种不同于以往文献中的证明技巧,给出了Euler方法均方稳定的一个充分条件.文末的数值试验证实了本文所获理论结果的正确性.     

非线性随机延迟微分方程Euler-Maruyama方法的均方稳定性

本文首先将数值方法的均方稳定性的概念MS-稳定与GMS-稳定从线性试验方程推广到一般非线性的情形,然后针对一维情形下的非线性随机延迟微分方程初值问题,证明了如果问题本身满足零解是均方渐近稳定的充分条件,那么当漂移项满足一定的限制条件时,Euler- Maruyama方法是MS-稳定的与带线性插值的Euler-Maruyama方法是GMS-稳定的理论结果．     

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods, the Milne–Simpson method and the BDF method. AMS subject classification (2000) 60H35, 65C30, 65L06, 65L20