随机延迟微分方程的数值方法的收敛性和稳定性

本文研究了随机延迟微分方程的平衡方法的收敛性和均方稳定性.利用半鞅收敛定理,给出了真解的渐进稳定和均方稳定的一个更弱的条件.平衡方法下随机延迟微分方程的真解的均方稳定性.     

随机延迟微分方程的数值方法的收敛性和稳定性（英文）

本文研究了随机延迟微分方程的平衡方法的收敛性和均方稳定性.利用半鞅收敛定理,给出了真解的渐进稳定和均方稳定的一个更弱的条件.平衡方法下随机延迟微分方程的真解的均方稳定性.     

中立型随机比例延迟微分方程平衡半隐式Euler方法的均方收敛性

本文讨论求解刚性中立型随机比例延迟微分方程的平衡半隐式Euler方法。证明了中立型随机比例延迟微分方程的平衡半隐式Euler方法是1／2阶均方收敛的。     

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.     

随机延迟微分方程Milstein方法的均方稳定性

本文以线性随机延迟微分方程为试验方程研究了随机延迟微分方程的Milstein方法的稳定性,给出了均方稳定的充分条件,所得结果表明Milstein方法能保持试验方程解的稳定性.完成了相关的数值试验以验证所得结论的正确性.     

Khasminskii型条件下随机延迟微分方程θ-方法的几乎必然指数稳定性（英文）

本文是我们之前工作的延伸,本文作者和殷荣城(2013)在单调型条件下考察了随机微分方程的θ方法的均方稳定性.在之前的结论中,我们考虑的是不带延迟的随机系统的均方稳定性.而本文,我们希望进一步考虑带延迟的随机系统的几乎必然稳定性.本文在修改后的Khasminskii条件下得到随机延迟微分方程θ方法的几乎必然指数稳定性.该结果使现有结论得到可观的推进.     

Khasminskii型条件下随机延迟微分方程$\theta$-方法的几乎必然指数稳定性[英文]

本文是我们之前工作的延伸, 本文作者和殷荣城(2013)在单调型条件下考察了随机微分方程的$\theta$ 方法的均方稳定性.在之前的结论中,我们考虑的是不带延迟的随机系统的均方稳定性.而本文, 我们希望进一步考虑带延迟的随机系统的几乎必然稳定性.本文在修改后的Khasminskii条件下得到随机延迟微分方程$\theta$方法的几乎必然指数稳定性. 该结果使现有结论得到可观的推进.     

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.     

非线性中立型随机延迟微分方程随机θ方法的稳定性

本文研究非线性中立型随机延迟微分方程随机θ方法的均方稳定性.在方程解析解均方稳定的条件下,证明了如下结论:当θ∈[0,1/2)时,随机θ方法对于适当小的时间步长是均方稳定的;当θ∈[1/2,1]时,随机θ方法对于任意步长都是均方稳定的.数值结果验证了所获结论的正确性.     

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.     

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.     

Exponential stability of Euler-Maruyama solutions for impulsive stochastic differential equations with delay

This paper establishes a method to study the exponential stability of Euler-Maruyama (EM) method for impulsive stochastic differential equations with delay. By using the properties of M-matrix and stochastic analysis technique, some conditions under which the EM solution is exponentially mean-square stable are obtained. Some examples are provided to illustrate the results.     

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.     

Mean-square exponential dichotomy of numerical solutions to stochastic differential equations

We present the ability of numerical simulations to reproduce the mean-square exponential dichotomy of stochastic differential equations. Under some conditions, we show that the mean-square exponential dichotomy of stochastic differential equations is equivalent to that of the numerical method for sufficient small step sizes     

Convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay

In this paper, we study the convergence and stability of the stochastic theta method (STM) for a class of index 1 stochastic delay differential algebraic equations. First, in the case of constrained mesh, i.e., the stepsize is a submultiple of the delay, it is proved that the method is strongly consistent and convergent with order 1/2 in the mean-square sense. Then, the result is further extended to the case of non-constrained mesh where we employ linear interpolation to approximate the delay argument. Later, under a sufficient condition for mean-square stability of the analytical solution, it is proved that, when the stepsizes are sufficiently small, the STM approximations reproduce the stability of the analytical solution. Finally, some numerical experiments are presented to illustrate the theoretical findings.     

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.     

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.