两类随机经济模型的动力学性质研究

发布时间：2018-03-11 10:33

本文选题：随机动力系统　切入点：经济模型　出处：《广西师范学院》2016年硕士论文　论文类型：学位论文

【摘要】：随着自然科学的不断发展,人们对现实世界的认识越来越贴近本质.因此,现实系统中不可避免的随机和非线性因素已成为众多数学家和其它领域科学家关注的焦点.特别是,近年来经济金融、生物系统等诸多领域中已推导出大量的非线性随机模型,进一步促使了人们对非线性随机微分动力系统的理论和应用的深入研究.为此,本文以随机分析和随机动力系统理论为工具,研究随机综合国力模型和一类非线性随机金融混沌系统的动力学性质.具体研究内容如下：第一章详细陈述了研究问题的背景及意义以及本文所要用到的一些预备知识,同时介绍了本学位论文研究的主要内容及其框架结构.第二章主要研究了随机综合国力模型的动力学性质.首先运用Poincare紧化变换将确定性综合国力模型的无穷远点转化为有限坐标点.进而,利用微分方程定性理论,研究其无穷远点的动力学行为,包括结点和鞍点的稳定性问题.其次,利用Lyapunov函数,鞅不等式和随机分析等方法和技巧,研究了Poincare紧化的随机综合国力模型的无穷远点的动力学性质,包括全局解的存在性,渐近矩估计,解的稳定性,矩均值,随机最终有界性.最后,研究了随机综合国力模型的轨道估计.第三章主要研究了一类非线性随机金融混沌系统的长期性行为.首先,利用截断函数方法和随机分析技巧,证明系统解的存在唯一性和有界性.其次,运用测度理论和Krylovs与Bogolyubov方法,证明系统存在平稳分布.最后,利用随机动力系统相关知识,证明系统存在唯一的随机吸引子,并通过数值模拟进行了验证.
[Abstract]:With the continuous development of natural science, people's understanding of the real world is more and more close to the essence. Therefore, the inevitable random and nonlinear factors in the real system have become the focus of attention of many mathematicians and scientists in other fields. In recent years, a large number of nonlinear stochastic models have been derived in many fields, such as economics, finance, biological systems, etc., which further promote the deep research on the theory and application of nonlinear stochastic differential dynamical systems. In this paper, stochastic analysis and stochastic dynamic system theory are used as tools. The dynamic properties of the stochastic comprehensive national strength model and a class of nonlinear stochastic financial chaotic systems are studied. The main contents are as follows: in chapter 1, the background and significance of the research and some preliminary knowledge to be used in this paper are described in detail. At the same time, the main contents and frame structure of this dissertation are introduced. In chapter 2, the dynamic properties of stochastic comprehensive national strength model are studied. Firstly, the infinity of deterministic comprehensive national strength model is transformed by Poincare compactness transformation. The point is transformed into a finite coordinate point. By using the qualitative theory of differential equations, the dynamical behavior of infinite points, including the stability of nodes and saddle points, is studied. Secondly, the methods and techniques of Lyapunov function, martingale inequality and stochastic analysis are used. In this paper, we study the dynamical properties of infinite points in Poincare compact stochastic comprehensive national strength model, including the existence of global solutions, asymptotic moment estimates, stability of solutions, mean moments, stochastic ultimate boundedness. In chapter 3, the long-term behavior of a class of nonlinear stochastic financial chaotic systems is studied. Firstly, the truncation function method and stochastic analysis technique are used. The existence, uniqueness and boundedness of the solution of the system are proved. Secondly, the existence of stationary distribution of the system is proved by means of measure theory and Krylovs and Bogolyubov methods. Finally, the existence of a unique random attractor is proved by using the relevant knowledge of the stochastic dynamical system. It is verified by numerical simulation.
【学位授予单位】：广西师范学院
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：F224

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