【作者】 王志勇；

【导师】 张诚坚；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2008， 博士

【摘要】 随机泛函微分方程可以看成是随机微分方程与确定性泛函微分方程的综合与推广,由于用该方程描述的系统既考虑了延迟因素又兼顾了随机扰动的影响,一般更能反映实际问题的需要,因而被广泛应用于物理、生物、机械、金融、控制等科学与工程领域的系统建模中。对随机微分方程,除少数线性情况外,一般的非线性方程都很难求其理论解,随机泛函微分方程更是如此,因而构造合适的数值算法对相应的解过程进行数值模拟就显得尤为重要了。相对于确定性问题,随机泛函微分方程由于随机因素的存在,在理论分析与算法构造的过程中,其难度和复杂性将大大提高。在随机泛函微分方程的理论中,稳定性分析是个重要课题。本文第二章针对一般的随机泛函微分方程,应用Razumikhin技巧,考察了理论解的p阶矩渐近稳定性,得到了一个非常一般的定理,并引发出许多相关的推论与结论,其中一个重要的应用就是可以推出Razumikhin-Mao定理的主要结果。在实际应用中体现了一定的价值。随机泛函微分方程是定义在连续函数空间上的一类方程,在实际计算中,只能考虑其特殊形式如随机延迟微分方程。随机延迟微分方程的数值处理是个新的研究领域,目前成果不多,主要是用Euler方法求解此类方程,并考察其线形均方稳定性。但Euler方法精度较低,只有强0.5阶。本文第三章将强1阶的Milstein方法应用于一般的线性随机延迟微分方程,得到其数值算法的一般格式,在分析了一类特殊的随机多重积分矩性质的前提下,考察了Milstein方法的均方稳定性,得到了当线性系统理论解满足均方稳定性条件下,对应数值解只需步长满足一定条件亦是均方稳定的理论结果。接着,在第四章我们将Milstein方法应用于一般的非线性随机延迟微分方程,得到了当理论解是均方稳定的,方程的漂移和扩散项满足一定的附加条件时,数值解也是均方稳定的结论。当前,关于随机延迟微分方程数值算法理论的研究举步维艰,其中一个重要的原因就是随机微分方程的数值解理论远不及确定性的成熟。在数值算法方面,比较系统的有随机Tayor方法,这类方法可以达到较高的精度,但是弊端是需要求高阶导数,比较麻烦。后来,人们研究了不含导数的随机Runge-Kutta方法,这方面的研究相当复杂与困难。本文第五章在Burrage,R(o|¨)ssler等人相关工作的基础上,研究了求解It(?)型随机微分方程强解的Runge-Kutta方法。构造了方法的一般格式,在应用发展随机Taylor展式和彩色树理论的基础上,得到了该类方法的阶条件,通过引入新的随机变量,得到了两类二级和3个三级的具体格式并分析了方法的均方稳定性。数值实验显示了所得方法良好的精度与广泛的适用性。当扩散项为零时,随机延迟微分方程即退化为确定性的方程。本文第六章将针对一类多延迟积分微分方程,考察扩展的一般线性方法的非线性稳定性,该方法的构造基于基本的一般线性方法和复合求积公式。理论分析证明了在适当条件下,该方法是渐近稳定和整体稳定的。更多还原

【Abstract】 Stochastic functional differential equations(SFDEs) can be viewed as generalizations of both deterministic functional differential equations(FDEs) and stochastic ordinary differential equations(SODEs).Since the random factor,together with the delay factor, is considred,SFDEs can always simulated the problems in practical truthfully.They have been widely used to model the corresponding systems in many scientific and engineering fields such as physics,biology,mechanics,finance and control.It is because analytical solutions are rare for stochastic differential equations that there is an increasing demand for numerical methods.As the existence of random factors,SDEs are more difficult and complex in constructing schemes rather than ordinary differential equations(ODEs).Among the researches on SFDEs,stability analysis is an important issue.At first,this paper deals with stability analysis of SFDEs in chapter 2.The p-th moment uniform asymptotic stability of the solutions is investigated by using Lyapunov functional and Razumikhin technique.A very general theorem is derived,by which we can get many corollaries,one of them is just the Ramumikhin-Mao theorem.As we all know,SFDEs are defined in the space of continuous functions.So only the solutions of the special functions such as stochastic delay differential equations(SDDEs) can be simulated.The investigation on numerical treatments of SDDEs is a new area.up to now,only few results have been presented.The main work is adapting the Euler-Maruyama method to solve the SDDEs,and studying the corresponding linear mean-square stability.But the convergence order of the method is only 0.5 and the accuracy is low.In chapter 3,we try to adapt the Milstein method with strong order 1 to solve the general linear SDDEs.and get the normal computational scheme,then the mean-square stability of the method is investigated by studying the moment property of a sort of stochastic multiple integrals. It is proved that the numerical method is mean-square stable under suitable conditions. The obtained result shows that the method preserves the stability property of the solved system.Furthermore,In chapter 4,the Milstein method is applied to nonliear SDDEs. When the analytical solution satisfies the conditions of mean-square stability,and if the drift term and diffusion term satisfy some restrictions,then the Milstein method is mean-square stable.At present,the research of numerical treatments fbr SDDEs is very difficult,one of the reasons is that the work in the area of SDEs is far less advanced than for deterministic differential equations.In the aspect of numurical schemes,the most common stochastic Taylor methods with derivatives of coefficient functions have been provided. to avoid finding the necessary derivatives,many derivative-free schemes such as Runge-Kutta(RK) methods are proposed.On the basis of the work by Burrage,R(o|¨)ssler,etc. In chapter 5,the paper constructs RK methods fbr strong solutions of the SDEs of It(?) type.We obtain the order conditions of the method by applying the colored rooted trees theory.Furthermore,we derive concrete schemes of two or three stages with order 1 by choosing new random variables.Furthermore,the mean-square stability of above methods is investigated.Finally,the accuracy of these methods is verified in numerical tests.When the diffusion term is equal to zero,SDDEs are degenerated to deterministic equations.In chapter 6,this paper deals with the nonlinear stability of extended general linear(GL) methods for solving multidelay integro- differential equations.The methods are composed of GL methods and compound quadrature rules.It is proved that the extended GL methods are asymptotically and globally stable under suitable conditions.更多还原