【作者】 王文强；

【导师】 李寿佛；

【作者基本信息】 湘潭大学 ， 计算数学， 2007， 博士

【摘要】 随机延迟微分方程既可以视为确定性模型问题延迟微分方程考虑了随机因素后的推广，也可以视为非确定性模型问题随机常微分方程考虑了时滞因素后的推广，所以随机延迟微分方程往往能够更加真实地模拟科学实际中的问题。因此它已开始被广泛地应用于物理、化学、控制论、金融学、神经网络、生态学等各个研究领域。由于它的研究方法，既不能等同延迟微分方程，也不能等同随机常微分方程，只会更加棘手，所以在具体的研究过程中必将会面临许多难以预料的困难。同时，与延迟微分方程和随机常微分方程一样，要想办法得到随机延迟微分方程问题本身的理论解是十分困难的。这就更加突显出随机延迟微分方程数值求解方法的研究工作是一件十分迫切而且具有极其重要意义的事情。随机延迟微分方程数值解法研究目前刚刚起步，国内外文献主要研究线性随机延迟微分方程数值方法．本文主旨是试图将这项研究推进到非线性情形，其次，目前文献中都是采用延迟量Υ是步长h的整数倍的技巧来处理问题，本文突破了这一局限利用插值方法来逼近延迟量．本文研究了几类求解非线性随机延迟微分方程的数值方法，获得了一系列收敛性与数值稳定性结果．所获结论可视为文献已有结果的推广．其中所获得的主要结果如下：(1)首先利用附近已有节点上的值通过线性插值的技巧对延迟项进行数值逼近，相对于现有文献中在延迟项的处理上，都是采用延迟量Υ是步长h的整数倍的技巧，这是一种崭新的尝试；然后针对较一般情形下的一类非线性随机延迟微分方程初值问题，得到了带线性插值的Euler-Maruyama方法在均方意义下的收敛性结果，它部分推广了已有文献中的相关结论。(2)将随机延迟微分方程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，得到了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。(3)将数值方法由Euler-Maruyama方法推广到更为广泛的一类方法——半隐式Euler方法(或称随机θ-方法)，同样我们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，得到了半隐式Euler方法是MS-稳定的与带线性插值的半隐式Euler方法是GMS-稳定的理论结果。(4)考虑到利用Milstein方法来求解一类极其重要的方程——Fokker-Planck方程，在此针对白噪声驱动随机系统的一维Fokker-Planck方程，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，那么当漂移项满足适度的限制条件时，获得了利用Milstein方法求解该方程时是MS-稳定的，而带线性插值的Milstein方法求解该方程时是GMS-稳定的理论结果。(5)考虑当扩散项为零时，非线性随机延迟微分方程就退化为确定性问题了，在此我们以非线性刚性变延迟微分方程为例，研究单支方法的D-收敛性，所获结论可以视为将文献中常延迟推广到变延迟的情形，证明了带线性插值的单支方法是p阶D-收敛的当且仅当该方法是A-稳定的且经典相容阶为p(这里p=1，2)。更多还原

【Abstract】 Stochastic Delay Differential Equations (SDDEs) can be regarded as extension of certain model problems for Delay Differential Equations (DDEs) which consider the random factor. It can also be regarded as the extension of uncertain model problems for Stochastic Ordinary Differential Equations (SODEs) considering the delay factor. Thus SDDEs can always simulate the scientific problem in practical truthfully. They have been widely used in Physics, Chemistry, Cybernetics, Finance, Neural Networks, Bionomics, etc. For the research method of SDDEs, it can neither be treated as DDEs nor SODEs, so it produces difficulties. The same as the DDEs and SDDEs, to obtain the theoretical solution of SDDEs will be very hard, hence it is important to study the numerical methods on solving the SDDEs.The study of numerical methods for solving SDDEs is underway at present, the research at home and abroad are mainly focused on linear SDDEs. In this paper, the main object is to extend the previous research results to nonlinear cases. Secondly, the delay termτis the multiple of stepsize h to dispose the problem, we approximate the delay term with linear interpolation to break this restriction. We study some kinds of numerical methods for solving nonlinear SDDEs, and obtain a series of stable and convergent results. These results can be regarded as the extension of the known results in present papers. The main results of this thesis are as follows:(1) First, we approximate the delay term with linear interpolation through the values of known neighbor points, this method is a new attempt corresponding with the known method for dealing with the delay term, which always use the technique entirely for treating the delay termτis the integer times of the stepsize h. For a kind of initial problems of nonlinear SDDEs, we then present the convergent results of Euler-Maruyama methods with linear interpolation procedure in the mean of mean-square. They are the extension of related results in previous papers.(2) The MS-stability and GMS-stability, which is the mean square concept of numerical methods for SDDEs, was extended from linear experiment equations to general nonlinear case. For one dimensional Initial Value Problem(IVP) of nonlinear SDDE, when the problem itself satisfies the sufficient condition that the zero solution is mean square asymptotically stable, we prove that when the drift term satisfies some given restrictions, the Euler-Maruyama methods is MS-stable and the Euler-Maruyama methods with linear interpolation procedure is GMS-stable.(3) We extend the Euler-Maruyama methods to a much more general class of semi-implicit methods, i.e. stochasticθ-methods. For the IVP of nonlinear SDDEs in one dimension, when the problem itself satisfies the sufficient condition that the zero solution is mean square asymptotically stable, we prove that when the drift term satisfies some given restrictions, the semi-implicit Euler method is MS-stable and the semi-implicit Euler method with linear interpolation procedure is GMS-stable.(4) Consider solving a class of very important equation, Fokker-Planck equations. For one dimensional Fokker-Planck equations of white noise driven by stochastic system, here we prove that the Milstein methods is MS-stable and the Milstein methods with linear interpolation procedure is GMS-stable for solving these problems when the problem itself satisfy the sufficient condition that the zero solution is mean square asymptotically stable.(5) When the diffusion term equals to zero, nonlinear SDDEs are degenerated to deterministic problems. This chapter is concerned with the error analysis of one-leg methods when applied to nonlinear stiff DDEs with a variable delay. It is proved that a one-leg methods with Lagrangian linear interpolation procedure is D-convergent of oder p if and only if it is A-stable and consistent of order p in the classical sense for ODEs. The results obtained can be regarded as extension of that for DDEs with constant delay presented in documents.更多还原