【作者】 谢颖；

【导师】 张诚坚；

【作者基本信息】 华中科技大学 ， 统计学， 2017， 博士

【摘要】 随机模型已经在很多的科学和工程领域的分支上起到了很重要的作用.越来越多的学者开始加入到研究随机微分方程的行列.随着近些年研究的深入,各种不同类型的随机微分方程开始获得学者的关注,比如,具常延迟的随机微分方程,具变延迟的随机微分方程,带泊松跳的随机微分方程,具马尔科夫转换的随机微分方程.可是,大部分带有延迟和其他类型随机过程的随机微分方程的精确解不能显式给出.因此,研究随机微分方程的数值解就显得越发的重要.本文针对几类具时变延迟的Ito型随机微分方程的解析解和数值算法进行了研究,着重研究了它们的数值收敛性,稳定性.在第一章,鉴于随机微分方程在各个领域的普遍应用,此文扼要的举出了若干个随机模型,回顾了随机微分方程以及其数值解研究的现实情况,介绍了本工作的主要内容和研究意义,并介绍了一些常用的记号、定义和基本理论.在第二章,我们考虑一类具分段常变元的非线性随机微分方程.利用随机单支θ-方法模拟此类随机微分方程.在全局Lipschitz和线性增长条件下,给出了该数值方法的收敛定理和收敛阶,此外还讨论了参数θ取不同值时,其数值解是否保持相应的指数稳定性.最后一部分利用数值试验证实了该方法的收敛阶和指数稳定性.第三章对于满足单边Lipschitz条件的具分段常变元的非线性带跳随机微分方程的数值解进行分析.我们选择了一类可以解决刚性问题的补偿分裂平衡法来处理这类随机微分方程.重点分析了此数值算法作用在此类方程上的强收敛性,分析当中我们用到了连续形式的数值格式而不是之前平衡法常用的离散形式的数值格式.并且,我们给出了此数值方法的数值解保持相应的解析解的指数稳定性所要满足的充分条件.章节的最后数值验证了补偿分裂平衡法的强收敛性和指数稳定性.在第四章,我们分析了作用在具马尔科夫调制的强非线性随机时变时滞微分方程的向后欧拉法.由于此类方程满足局部Lipschitz条件和单边多项式增长条件,它具有很强的非线性,此外还受到时变延迟的影响,因此分析这类方程的数值解时往往很难得到其强收敛的性质.为此我们引入了几个引理,在证明的过程中加上了处理时变延迟的技巧,证明了向后欧拉法作用在此类方程的强收敛性.此外,利用连续型和离散型的半鞅收敛理论,证明了其数值解在满足一定的条件下保持方程解析解的几乎必然指数稳定性.我们给出了相应的数值试验来验证我们的理论.在第五章,针对一类具马尔科夫调制的强非线性随机中立型微分方程,我们证明了此类方程的渐近有界性和P次指数稳定性.文中分别给出了两个主要定理,我们基于李雅普诺夫理论分析得到了第一个定理,依据第一个定理和M-矩阵的性质,第二个定理给出了当系数项满足某些条件时,方程是渐近有界和p次指数稳定的.在这样的分析下,可以发现,这类方程即使有某个状态不是渐近有界或指数稳定的,依然可能保持整体方程在无限时间上的渐近有界性或指数稳定性.最后,一个数值例子验证了我们的结论.最后一章,我们对本工作进行了总结,对以后的工作进行了展望.更多还原

【Abstract】 Stochastic modelling has come to play an important role in many branches of science and engineering.More and more scholars begin to study the stochastic differential equation.With the deepening of the research in recent years,various classes of stochastic differential equations began to get the attention of scholars,for example,stochastic differential equations with constant delay,stochastic differential equations with time-variable delay,poisson jump-diffusion stochastic differential equations,stochastic differential equations with Maikovian switching.However,the exact solutions of most stochastic differential equations with delays and other types of random process cannot be given explicitly.Thus,it’s becoming increas-ingly important to research the numerical solution of stochastic differential equation.In this work,we consider the analytic solutions and the numerical algorithms of several classes of Ito stochastic differential equations with time-variable delay and focus on studying the convergence and stability.In Chapter 1,in view of the stochastic differential equation widely used in various fields,we briefly cite several stochastic models.Morevoer,we review the research develop-ment of stochastic differential equations and its numerical algorithm.Also,we introduce the main content and the research significance of this work,and introduce some commonly used notations,definitions and basic theory.In Chapter 2,we consider nonlinear stochastic functional differential equations with piecewise continuous arguments(SFDEPCAs).The underlying one-leg θ-methods are adapted to solve SFDEPCAs.Under the global Lipschitz condition and the strongly linear growth condition,it is proved that the adapted stochastic one-leg θ-methods are convergent with strong order 1/2.Moreover,we give a mean-square exponential stability criterion of the adapted one-leg θ-methods.At last,some numerical experiments are given to illustrate the computational effectiveness and the theoretical results of the induced methods.Chapter 3 deals with numerical solutions for nonlinear stochastic differential equa-tions with jump-diffusion and piecewise continuous arguments under the one-sided Lips-chitz condition.The compensated split-step balanced methods(CSSBB)for the equations are suggested.It is proved that the solutions of CSSBB methods are convergent using the continuous-time approximation rather then discrete approximation which is usually ana-lyzed in a class of balanced methods.Also,we give the criterion that when the CSSBB methods satisfy the corresponding conditions,the mean-square exponentially stability of the numerical solutions for CSSBB methods preserves the mean-square exponentially sta-bility of the analytical solutions.At last,two numerical examples show that CSSBB methods are more effective than those methods which are without balanced term.In Chapter 4,we focus on the backward Euler-Maruyama method for nonlinear hybrid SDEs with time-variable delay.Under the local Lipschitz condition and polynomial growth condition,the equation has a highly nonlinear problem and the effect of time-varying delay.Therefore,it’s always difficult to get the strong convergence properties of the numerical solution.By citing several lemmas and the technique for dealing with time-varying delay,it is proved that the backward Euler-Maruyama method is strongly convergent.Moreover,applying the continuous and discrete semi-martingale convergence theorems proves that nu-merical methods preserve the almost surely exponential stability of the analytic solutions to highly nonlinear hybrid SDEs with time-variable delay under the appropriate conditions.A numerical experiment is given to illustrate the computational effectiveness and the theoreti-cal results of the method.Chapter 5 is concerned with asymptotical boundedness and moment exponential sta-bility for nonlinear stochastic neutral differential equations with time-variable delay and markovian switching.Two main theorems are given respectively.First,we give the analysis of the Lyapunov theory for nonlinear hybrid neutral stochastic differential delay equations.Based on the first criterion and the property of M-matrices,the second criterion shows the asymptotic boundedness and the moment exponential stability are tenable when we give the general conditions depended on coefficients of the nonlinear hybrid neutral stochastic dif-ferential delay equations.Under the analysis,we can find that even if there is a certain state not asymptotically bounded or exponentially stable,the systems still keep asymptotically bounded or exponentially stable.At last,an example is also given to illustrate the theoretical results.In the last chapter,this work is summarized,and we describe the prospects for the future.更多还原