随机微分方程截断Euler-Maruyama方法的几乎处处稳定性法

发布时间：2018-02-14 09:04

本文关键词： 随机微分方程截断Euler-Maruyama 半鞅收敛定理几乎处处稳定超线性增长　出处：《上海师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：全局Lipschitz条件下,随机微分方程的数值格式的收敛性和稳定性已经得到了很好的研究,但实际上只有少部分随机微分方程的系数都满足全局Lipschitz条件,因此研究弱化全局Lipschitz条件下随机微分方程的数值解是有必要的。2002年Highm,Mao和Stuart合作发表的在局部Lipschitz和线性增长条件下研究随机微分方程数值解强收敛性的结果为研究非Lipschitz条件下随机微分方程数值解开辟了一条新途径。但是,线性增长条件仍然过于严格,当随机微分方程的漂移系数和扩散系数为超线性时,仍可能无法通过已有的数值格式对原问题进行分析与求解。而且,有研究结果表明显式Euler-Maruyama(EM)方法在处理超线性随机微分方程的时候,不能保证其收敛性。隐式方法可以被看做处理这类问题的可行方案,但隐格式计算的复杂性和成本都较高,从计算复杂度、格式的简单程度等几个方面来看,显式方法仍然具有优势。因此,近期较多成果着重于改进经典显式EM方法来处理超线性随机微分方程以保证数值格式的收敛性和稳定性,相应的格式主要有驯服(tamed)EM方法,stopping EM方法以及截断(truncated)EM方法等等,本文我们重点关注截断EM方法。只有充分研究了数值格式的收敛性与稳定性,该格式才是可用的。数值格式的收敛性是不可忽略的一个环节,毛学荣教授在2015和2016年两篇关于截断EM方法的文章中相继证明了其强收敛性以及估计出了其强收敛阶。但是目前为止尚未有关截断EM方法稳定性的分析,本文主要研究局部Lipschitz与Khasminskii等条件下的随机微分方程以及随机时滞微分方程的截断EM方法的稳定性。非线性稳定性主要有Lp与几乎处处稳定两种,本文我们将主要运用半鞅收敛定理研究几乎处处稳定性。我们还给出了部分数值例子进行验证。
[Abstract]:Under the global Lipschitz condition, the convergence and stability of the numerical schemes of stochastic differential equations have been well studied, but in fact, only a few of the coefficients of stochastic differential equations satisfy the global Lipschitz condition. Therefore, it is necessary to study the numerical solutions of stochastic differential equations under the condition of weakening global Lipschitz. In 2002, the results of the study on the strong convergence of numerical solutions of stochastic differential equations under the condition of local Lipschitz and linear growth were published by Highm Mao and Stuart in 2002. The numerical solution of stochastic differential equations under non-#en0# conditions opens up a new way. The linear growth condition is still too strict. When the drift coefficient and diffusion coefficient of the stochastic differential equation are superlinear, it may not be possible to analyze and solve the original problem by the existing numerical scheme. Some research results show that the explicit Euler-Maruyamaan (EMEM) method can not guarantee the convergence of superlinear stochastic differential equations. The implicit method can be regarded as a feasible scheme to deal with this kind of problems, but the computational complexity and cost of implicit schemes are higher. In terms of computational complexity, simplicity of the format, and so on, explicit methods still have advantages. More recent achievements have focused on improving the classical explicit EM method to deal with superlinear stochastic differential equations to ensure the convergence and stability of numerical schemes. The corresponding schemes include tame tamedonian EM method and truncated uncatered EM method, etc. In this paper, we focus on truncated EM method. Only when the convergence and stability of numerical schemes are fully studied, can the scheme be used. The convergence of numerical schemes is a link that can not be ignored. Professor Mao Xuelong has proved its strong convergence and estimated its strong convergence degree in two articles about truncated EM method in 2015 and 2016. However, there is no analysis on the stability of truncated EM method so far. In this paper, the stability of stochastic differential equations under local Lipschitz and Khasminskii conditions and the truncated EM method of stochastic delay differential equations are studied. There are two kinds of nonlinear stability: LP and almost everywhere stability. In this paper, we will mainly use the semimartingale convergence theorem to study almost everywhere stability, and we also give some numerical examples to verify it.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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