Modelling And Analysis Of Financial Fluctuations By Stochastic Interacting Systems

By applying the theory of continuum percolation, interacting systems and statis-tical physics, Ising model, voter model and Zipf plot method, we mainly discuss the statistical properties of fluctuations of the stock price process, the properties of the in-teracting reaction of two stock indices, and the fat tails phenomena and the power law distributions of Shanghai Stock Exchange Index and Shenzhen Stock Exchange Index. We discuss the corresponding valuation and hedging of European contingent claims for the price process model. And we also investigate the statistical behaviors of two-layered random phase interfaces in two-dimensional W-R model. This thesis is organized as follows.In Chapter2, we investigate the statistical properties of fluctuations of the stock price process in a stock market by continuum percolation theory. The methods of con-tinuum percolation are applied to construct a financial model that describes the behavior of a stock price, specifically the continuum percolation is used to describe the "herd effect" of investors in a financial market. By using the stochastic methods of statistical analysis, we show that the characteristic function of this stock price process converges to the corresponding characteristic function of Black-Scholes model.In Chapter3, we consider the statistical properties of chain reaction of stock in-dices. The theory of interacting systems and statistical physics are applied to describe and study the fluctuations of two stock indices in a stock market, and the properties of the interacting reaction of the two indices are investigated in the present paper. In this work, stochastic analysis and the two random paths model are used to study the proba-bility distribution for the chain reaction of stock indices, further we show the asymptoti-cal behavior of probability measures of the fluctuations for the two stock indices model and the probability properties of one stock index by chain reaction. We discuss the convergence of the finite dimensional probability distributions for the financial model.In Chapter4, by applying the theory of stochastic processes and interacting parti-cle systems and models, including stopping time theory and stochastic voter model, we model a financial stock price model that contains two types of investors. And we use this financial model to describe the behavior and fluctuations of a stock price process in a stock market. In the financial model, besides the professional investors, we also consider the general investors. Where the stopping time and the voter model are ap- plied to model and study the statistical properties of investment of the general investors. Further, we discuss the valuation and hedging of European contingent claims for this price process model.In Chapter5, the statistical behaviors of two-layered random phase interfaces in two-dimensional Widom-Rowlinson model are investigated. The phase interfaces sep-arate two coexisting phases of the lattice Widom-Rowlinson model, when the chemical potential ? of the model is large enough, the convergence of the probability distribu-tions which describe the fluctuations of the phase interfaces is studied. The backbones of interfaces are introduced in the model, and the corresponding polymer chains and cluster expansions are developed and analyzed for the polymer weights. And the exis-tence of the free energy for two-layered random phase interfaces of the two-dimensional Widom-Rowlinson model is given.In Chapter6, the fluctuations of stock prices and trade volumes are investigated by the method of Zipf plot, where Zipf plot technique is frequently used in physics science. In the first part, the data of stock prices and trade volumes in Shanghai Stock Exchange and Shenzhen Stock Exchange is analyzed, the statistical properties of stock prices and trade volumes are studied. We select the daily data for Chinese stock market during the years2002-2006, by analyzing the data, we discuss the statistical properties of fat tails phenomena and the power law distributions for the daily stock prices and trade volumes. In the second part, we consider the fat tails phenomena and the power law distributions of Shanghai Stock Exchange Index and Shenzhen Stock Exchange Index during the years2001-2006by Zipf plot method.In Chapter7, some mathematical finance questions related to this thesis are listed.