Mathematical Basis Of The Fractal-dimension And Empirical Analysis Of The Chaos And Fractal To Shanghai Stock Market

The theory of chaos and fractal have are widely applied on economics and finance field since the 70's last century. Talking about our country's studies on this way, as whole, these studies as followed have been doing, recognizing of system chaos, looking for chaos attractors, researching fractal structure to time series curve, prediction and control to chaos system etc. All those studies need deal with the estimation of the fractal-dimension. There are many definitions and estimation methods of the fractal-dimension in practical applied. Therefore, it is necessary to discuss the mathematical basis and calculation methods of the fractal-dimension and recognizing the definition and mutual correlation of many useful fractal-dimension.The chapter one of this paper we introduce the application of chaos and fractal in economics and finance, give out the knowledge of the correlation to chaos and fractal, and point out that the estimation of the fractal-dimension is very important in research. In chapter two and chapter three, we discuss in some detail the mathematics theory and estimate methods of the fractal-dimension, including the mathematical definition and applied characters of almost all the fractal-dimension. Specially, we give out estimation manners of correlation-dimension and Hurst exponent that are widely used while calculating the fractal-dimension of a time series. At last, through estimating the correlation-dimension and Hurst exponent to Shanghai Stock Index, we give some empirical analysis about the characters of chaos and fractal on Shanghai stock market by using SAS and MATLAB. Computation results on real data from Shanghai stock market obviously show that the fractal-dimension on Shanghai stock market is approximately 1.5. And furthermore, Shanghai stock market exists chaos attractor.