分数维动态系统分析理论及其在中国证券市场中的应用

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摘要

本论文的研究工作得到国家自然科学基金项目（2004.1－2006.12 编号

70371070）“复杂系统结构演化边界带方法研究及其在金融系统管理中的应用”的

支持。

本文基于分形市场理论，以更好地研究具有分形结构的资本市场的行为和规

律，提出分数维动态系统分析方法，并对沪深股市进行了实证研究。

本文的主要研究工作如下：

1.分析了有效市场理论的缺陷，对分形市场研究的发展、现状做了阐述，并

对国内外资本市场的非线性研究进行了评述。

2.综合考察了分数维系统的特性再结合复杂系统的共性，提出了分数维动态

系统分析方法，分数维动态系统主要由三方面构成：一是系统的分数维性；二是

维度的结构性；三是系统、边界和环境的动态变化性。

3. 明确系统边界带要素对系统演化的重要性。系统边界带与系统和环境的简

单结合处是有区别的，系统边界带通常以变换形态存在，在系统演化的过程中起

了很重要的作用。

4.研究系统和边界带、环境之间的动态行为,明确了系统向环境游离的能量、

物质、信息与其相应的捕获之间的平衡对系统稳定性（结构和功能）的影响。

5.对资本市场分形结构的判断，用到分形理论中的 R/S 分析方法、BDS 检验、

ARIFMA 模型。得出 98 年以后，中国股市的长期记忆效应或持久性特征趋于弱化，

表现为短期记忆效应，表明中国证券市场渐进趋于弱势有效的结论。

6.建立体现分数维动态系统分析方法的动态模型。利用分析结果确立系统要

素的关系及其动态变化和阶段作用性。根据股市系统、边界带和环境综合模型，

用计量经济学的方法对证券市场的稳定性进行检验，用 R/S 分析的方法，计算赫

斯特指数对维度的变化做出分析，综合沪深股市指数的表现维、涨跌维、分数维

进行分析，用系统动力学的方法对系统、环境和边界带之间的动态过程进行分析。

建立基于分数维系统分析理论的股票市场系统模拟，反映系统边界带与系统间关

系和系统边界带与环境间关系等动态关系。

7.对股市系统进行模拟，得出结果。将上述结果综合并做整体处理。采用上

证指数的数据，进行模拟，并得出模拟结果。

本文创新点：

1.本文针对具有分数维结构的系统进行研究，采用理论与实际相结合的方法，

对分数维动态系统进行研究。这几十年里国内外学者专注于系统分形性的诊断，

是一种定性分析，而没有形成专用于研究分数维结构系统的分析方法。

2.对 R/S 分析方法进行改进，采用非线性方法估计

指数避免纯粹为了简化

计算的线性转化过程会导致有偏的参数估计。

3.基于维度对复杂系统外观形式的描述，考虑边界带因素的影响，结合系统

与环境之间的信息博弈，根据系统的演进、演退性来对分数维动态系统进行分析，

并给出演化轨迹的预测。

4.研究了分数维动态系统中系统、环境和边界带之间物质、能量和信息的游

离与捕获。丰富了复杂系统结构演化边界带方法，将复杂系统的研究推进了一步。

本文的研究工作提出分数维动态系统分析理论来对复杂系统进行进一步的研

究，其成果将为研究具有分数维的系统提供一个新的视角，具有较大的理论意义

和实际应用价值。

关键词：分数维动态系统分形信息博弈边界带维度

ABSTRACT

This paper is based on the fractal market hypothesis theory, in order to better study

the capital market behavior and laws with fractal structure, put forward the fractal

dimensional dynamical system analysis theory. In addition, we applied the theory into

Shanghai and Shenzhen stock markets.

As a part of the National Natural Science Foundation of China (No.70371070), this

paper is studied with ten chapters. Summarizing the entire dissertation, the paper’s

primary research job is as follows:

1. Analyse the limitation of EMH, and set forth the develpment, actuality of FMH,

at last narrate the research of non-linear of captal market.

2. Mainly research for the fractal dimensional dynamical system analysis theory,

put forward the definition of the fractal dimensional dynamical system and on the basis

of that think the four factors of the fractal dimensional dynamical system theory include:

the evolution of dimension, information game, forward-evolvibility and

backward-evolvibility, and the boundary strip behavior analysis of the fractal

dimensional dynamical system analysis theory. And these four elements are respectively

analyzed studies.

3. Define the essentiality of system boundary clearly. It should be noted that

system boundary, is different than the simple joint existing between system and

environment. The system boundary always exists in a transformation form, and it cannot

be replaced by the simple concept of joint between system and environment. Define

clearly the dimension of system boundary is helpful for us to the further understanding

of the dimension relationship between system and environment.

4.Resarch for the dynamic action among system, boundary and environment, and

explain which balance relation between the dissociation of energy, matter and

information from system to enviorment and the capture of that is importmant for the

stability of system (structure and function). Vice versa, the stability of system has great

significance to that, which is the necessary precondition of system applying function.

5. The stock market nonlinear characteristics certification study, which mainly in

this section, analyze the characteristics of capital markets fractal, and take the stock

market for example, whose nonlinear features were certified, attain the result there exist

the long-term memory characteristic in the Shanghai and Shenzhen stock market. 98

years later, the stock market effects of long-term memory or durability characteristics

tend to weaken, and the performance of short-term memory appear, which indicate

China's securities market has shown gradual vulnerable effective.

6. Positive application researches of FDDSAT basing on Chinese stock market

(mainly Shanghai Stock Exchange market). The contents mainly include dimensional

positive applications researches (including Behaving dimension, Rising -falling

dimension, Fractal dimension, Progressional dimension of Shanghai stock exchange

indexes and their compositive analyses), In accordance with the plan (the stock market

system, border belt and environment integrated models) scores based on the theoretical

analysis of the Victoria stock market simulation system, which fully reflected system,

the environment, and the border belt system focus structures and systems and the

relationship between the environment (total relations), system and the relationship

between the internal elements, the internal relations between elements, elements within

the border belt with the relationship between, border belt system and the relationship

between the system and the border belt system with the dynamics of the relationship

between environmental groups.

7. Synthesize the above results and deal with them integrally. The final results

includes not only the quantification of the items in the 5th step, but also which

relationship is overwhelming during some period, when and which elements have the

most affect on the system function, and when the dimension of system would change.

The paper’s innovation point is as follows:

1.In the paper mainly research for the fractal dimensional system, by the method of

theory conjoint with prctice. For these decades scholars have paied much attation to

diagnose the fractal characteristic of system, which is one kind of qualitative nanlysis,

don’t find an analysis method to research for the system with fractal dimension.

2. Improve on the R/S analysis method. Using the non-linear method estimate H

exponet in order to avoid defletion of parameter estimation resulted from simply

calculate.

3. Based on the depiction of dimension to the appearance of complex system,

consider the infection of boundary factor, combine the information game between the

system and enviroment, according to the evolvement go ahead and back off of system

analyse the fractal dynamic system, and forecast the track of evolvement.

4. Research for the dissociation and captur of matter, energy and information

among system, enviorment and boundary in the fractal dynamic system.

As we know, in this paper raising the fractal dimensional dynamic system analysis

theory further study complex system, which offer a new research angle to system with

fractal dimension, provide a perspective and a positive stock market in China, with the

larger theoretical and practical application value.

Key words: the Fractal dimensional dynamic system, Dimension,

Information game, Boundary, Dimension

中文摘要

ABSTRACT

第一章绪论...................................................................................................................1

§1.1 论文研究的背景与研究意义 .............................................................................. 1

§1.1.1 本文选题的背景............................................................................................1

§1.1.2 研究意义........................................................................................................2

§1.2 经济系统复杂性研究的发展及其评述 .............................................................. 4

§1.2.1 资本市场复杂性国内外研究概况................................................................4

§1.2.2 目前复杂性研究的评述................................................................................9

§1.2.3 分形市场边界带的想法..............................................................................11

§1.3 分数维动态系统的分析方法 ............................................................................ 13

§1.3.1 分数维动态系统的定义..............................................................................14

§1.3.2 四个基础问题..............................................................................................14

§1.4 论文的结构 ........................................................................................................ 17

§1.5 本章小结 ............................................................................................................ 19

第二章分形市场理论...................................................................................................21

§2.1 有效市场理论的发展和局限性 ........................................................................ 21

§2.1.1 有效市场理论发展概述..............................................................................21

§2.1.2 有效市场的局限性......................................................................................23

§2.2 分形市场研究及评述 ........................................................................................ 25

§2.2.1 非线性经济学的兴起..................................................................................25

§2.2.2 资本市场的分形市场研究..........................................................................26

§2.2.3 对分形市场研究的评述..............................................................................31

§2.3 分形市场理论的特征 ........................................................................................ 32

§2.3.1 金融市场的非线性......................................................................................32

§2.3.2 金融市场的正反馈机制..............................................................................34

§2.3.3 金融波动的长记忆性..................................................................................34

§2.3.4 分形市场理论的意义..................................................................................35

§2.3.5 中国股票市场的非线性特征验证研究现状及评述..................................37

§2.4 本章小结 ............................................................................................................ 38

第三章股票市场的分形特征验证...............................................................................39

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摘要：

摘要本论文的研究工作得到国家自然科学基金项目（2004.1－2006.12编号70371070）“复杂系统结构演化边界带方法研究及其在金融系统管理中的应用”的支持。本文基于分形市场理论，以更好地研究具有分形结构的资本市场的行为和规律，提出分数维动态系统分析方法，并对沪深股市进行了实证研究。本文的主要研究工作如下：1.分析了有效市场理论的缺陷，对分形市场研究的发展、现状做了阐述，并对国内外资本市场的非线性研究进行了评述。2.综合考察了分数维系统的特性再结合复杂系统的共性，提出了分数维动态系统分析方法，分数维动态系统主要由三方面构成：一是系统的分数维性；二是维度的结构性；三是系统、边界和环境的动态...

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