求解非线性互补问题的微分方程方法

本文构造了一种求解非线性互补问题的微分方程方法.在一定条件下,证明了微分方程系统的平衡点是非线性互补问题的解并且基于一般微分方程系统的数值积分建立了一个数值算法.在适当的条件下,证明了此算法产生的序列解是收敛的.本文最后给出了数值结果,该结果表明了此微分方程方法的有效性.     

多维马尔科夫转制随机微分方程的数值解

由于多维马尔科夫转制随机微分方程不存在解析解,利用Euler—Maruyama方法给出多维马尔科夫转制随机微分方程的渐进数值解,并证明了此数值解收敛到方程的解析解．将单一马尔科夫转制随机微分方程的数值解问题延伸到多维马尔科夫转制情形,增强了马尔科夫转制随机微分方程的适用性．     

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.     

向前型分段连续微分方程的数值解

本文讨论了向前型分段连续微分方程Euler-Maclaurin方法的收敛性和稳定性,给出了Euler-Maclaurin方法的稳定条件,证明了方法的收敛阶是2n+2,并且得到了数值解稳定区域包含解析解稳定区域的条件,最后给出了一些数值例子用以验证本文结论的正确性.     

拟等级网格下非线性延迟微分方程间断有限元法

本文利用间断有限元法求解非线性延迟微分方程,在拟等级网格下.给出非线性延迟微分方程间断有限元解的整体收敛阶和局部超收敛阶,数值实验验证了理论结果的正确性.     

一类随机泛函微分方程带随机步长的EM逼近的渐近稳定

研究了一类带有限延迟的随机泛函微分方程的Euler-Maruyama(EM)逼近，给出了该方程的带随机步长的EM算法，得到了随机步长的两个特点:首先，有限个步长求和是停时；其次，可列无限多个步长求和是发散的.最终，由离散形式的非负半鞅收敛定理，得到了在系数满足局部Lipschitz条件和单调条件下，带随机步长的EM数值解几乎处处收敛到0.该文拓展了2017年毛学荣关于无延迟的随机微分方程带随机步长EM数值解的结果.     

非Lipschitz系数McKean-Vlasov随机微分方程的大偏差原理

本文研究有限维框架下一类非Lipschitz系数McKean-Vlasov随机微分方程的Freidlin-Wentzell型大偏差原理,将此类条件下经典随机微分方程的相关结论推广到McKean-Vlasov随机微分方程.在此类McKean-Vlasov随机微分方程解的存在唯一性基础上,采用弱收敛方法得到其大偏差原理.     

带有Poisson随机测度的比例微分方程数值解的收敛性

主要研究了带跳的随机比例微分方程dX(t)=f((X(t),X(qt))dt+g(X(t),X(qt))dW(t)+∫nh(X(t),X(qt),u)N(dt,du),0≤t≤T,X(0)=X0,给出了此方程的Euler数值解,并在局部Lipschitzs条件下,证明了数值解依均方和概率测度意义下收敛于精确解.     

精确有限元法

本文提出构造有限单元的新方法——精确有限元法.它可以求解在任意边界条件下任意变系数正定或非正定偏微分方程。文中给出它的收敛性证明和计算偏微分方程的一般格式。用精确元法所得到的单元是一个非协调元,单元之间的相容条件容易处理.与相同自由度普通有限元相比,由精确元法所得到的解的高阶导数具有较高的收敛精度.文末给出数值算例,所得到的结果均收敛于精确解,并有较好的数值精度.     

自共轭常微分方程奇异摄动问题一致收敛的差分格式

本文对自共轭常微分方程奇异摄动问题,构造一族带拟合因子的差分格式,用不同于[1]的方法,通过对格式截断误差的分析,给出差分格式解一致收敛于微分方程解的充分条件;由此提出几个具体的差分格式,在较弱的条件下,给出较高的一致收敛阶,并将它们应用于例子,给出数值结果.     

带有泊松跳跃马尔可夫调制的中立型随机时滞微分方程近似解的依概率收敛

本文在局部Lipschitz条件和一些附加条件下得到了方程的全局解, 而未使用线性增长条件. 另外, 对带有泊松跳跃马尔可夫调制的中立型随机时滞微分方程近似解的收敛性进行了研究, 取代了以往的均方收敛方式, 改为依概率收敛. 从而对现有的一些结果进行了改进.     

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Stability in distribution of stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems. There are several papers which study this stability for stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching technically. In our paper, we are concerned with the general neutral stochastic functional differential equations with Markovian switching and we derive the sufficient conditions for stability in distribution. At the end of our paper, one example is established to illustrate the theory of our work.     

马尔可夫调制及带Poisson跳随机时滞微分方程依分布稳定

本文讨论马尔可夫调制及带Poisson跳随机时滞微分方程,其主要目的是研究方程解的依分布稳定.     

Convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching

In this paper, a class of stochastic age-dependent population equations with Markovian switching is considered. The main aim of this paper is to investigate the convergence of the numerical approximation of stochastic age-dependent population equations with Markovian switching. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions. An example is given for illustration.     

Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations

In this paper we study the mean-square (MS) stability of the Milstein method for linear stochastic delay integro-differential equations (SDIDE) with Markovian switching by extending the techniques of [Z. Wang, C. Zhang, An analysis of stability of Milstein method for stochastic differential equations with delay, Computers and Mathematics with Applications 51 (2006) 1445–1452; L. Ronghua, H. Yingmin, Convergence and stability of numerical solutions to SDDEs with Markovian switching, Applied Mathematics and Computation 175 (2006) 1080–1091]. It is established that the Milstein method is MS-stable for linear stochastic delay differential equations (Wang and Zhang (2006); in the above reference). Here we prove that it is MS-stable for linear SDIDE with Markovian switching also under suitable conditions on the integral term. A numerical example is provided to illustrate the theoretical results.     

Convergence of the semi-implicit Euler method for neutral stochastic delay differential equations with phase semi-Markovian switching

Recently, in the numerical analysis for stochastic differential equations (SDEs), it is a new topic to study the numerical schemes of neutral stochastic functional differential equations (NSFDEs) (see Wu and Mao [1]). Especially when Markovian switchings are taken into consideration, these problems will be more complicated. Although Zhou and Wu [2] develop a numerical scheme to neutral stochastic delay differential equations with Markovian switching (short for NSDDEwMSs), their method belongs to explicit Euler–Maruyama methods which are in general much less accurate in approximation than their implicit or semi-implicit counterparts. Therefore, to propose an implicit method becomes imperative to fill the gap. In this paper we will extend Zhou and Wu [2] to the case of the semi-implicit Euler–Maruyama methods and equations with phase semi-Markovian switching rather than Markovian switching. The employment of phase semi-Markovian chains can avoid the restriction of the negative exponential distribution of the sojourn time at a state. We prove the semi-implicit Euler solution will converge to the exact solution to NSDDEwMS under local Lipschitz condition. More precise inequalities and new techniques are put forward to overcome the difficulties for the existence of the neutral part.     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

Stability and stochastic stabilization of numerical solutions of regime-switching jump diffusion systems

This work studies stability and stochastic stabilization of numerical solutions of a class of regime-switching jump diffusion systems. These systems have a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering and economics because they involve three classes of stochastic factors: white noise, Poisson jump and Markovian switching. This paper focuses on the stability of numerical solutions of the switching jump diffusion systems and examines the conditions under which the Euler–Maruyama (EM) and the backward EM may share the stability of the exact solution. These conditions show that all these three classes of stochastic factors may serve as stabilizing factors and play positive roles for the stability property of both exact and numerical solutions.     

An analytic approximation of solutions of stochastic differential delay equations with Markovian switching

In this paper, we are concerned with the stochastic differential delay equations with Markovian switching (SDDEwMSs). As stochastic differential equations with Markovian switching (SDEwMSs), most SDDEwMSs cannot be solved explicitly. Therefore, numerical solutions, such as EM method, stochastic Theta method, Split-Step Backward Euler method and Caratheodory’s approximations, have become an important issue in the study of SDDEwMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEwMSs in the sense of the L^p-norm when the drift and diffusion coefficients are Taylor approximations.