| The modeling of financial market price fluctuation and the corresponding nonlinear statistical analysis is one of focuses in the field of financial mathematics and financial engineering.The research on the model will help people better understand the internal mechanism and characteristics of financial market,and play a key role in financial risk management,physical asset evaluation and derivative pricing.Financial market is a complex interaction system,in recent years,a large number of financial price models based on stochastic interacting particle systems have been proposed gradually.These models reproduce and study the nonlinear dynamics of financial market by focusing on the behavior of individual participants in order to reveal the evolution law of complex financial market systems.In this paper,several important stochastic interacting particle systems are applied to the modeling of financial market price fluctuation.Combined with the stochastic theory of compound Poisson process,three new Lévy type financial stochastic interacting system models are established.This paper investigates the possible formation dynamics of price volatility from the micro and macro perspectives.This paper studies the stylized facts and nonlinearity of the simulated return series and the real return series from multiple perspectives and multi-dimensional parameters.The research methods include statistical analysis,nonlinear analysis and complexity analysis.The abundant empirical results show that these three Lévy type financial stochastic interacting system models are reasonable.In Chapter 1,we introduce the topic background,research situation,theoretical basis,and main innovations of this paper.In Chapter 2,a stochastic interacting financial system model with jumps based on stochastic Ising model and continuous percolation system is studied to understand the price fluctuation of financial market.The micro mechanism of the interaction between traders in the model is realized by the interaction of particles(positive spin and negative spin)in Ising dynamic system,that is,having positive position “+” or negative position“-” in financial market.The position of a trader is affected by the market information,and the positive and negative proportion of trading positions will lead to the rise and fall of the price for the financial asset.In addition,the information of sudden changes in the external investment environment or internal operation state will lead to sharp jump fluctuations in the actual financial market.The herd behavior of traders is also an important factor that leads to the drastic fluctuation of financial market.In the model,the continuous percolation system combined with Poisson process is used to reproduce these sudden jumps.The range of the jump of the financial asset is simulated by the clusters of local interaction in the continuous percolation system.By the statistical analysis and complexity analysis,this paper studies the volatility behavior and various complex properties of the returns for the proposed model and the real market,and introduces p-order multi-scale autocorrelation function and new q-order multi-scale entropy to scale analyzing the simulated data and the real data,in order to verify the effectiveness of the model.In Chapter 3,a new financial stochastic interacting price model is established by Potts model and oriented percolation system.The micro mechanism of the interaction among traders is realized by the interaction of particles(spins)in Potts dynamic system.The parallel spin string in lattice Potts model is defined as the investor group with the same operation behavior(or the same investment attitude)in the market,The oriented percolation system combining with Poisson process is used to realize the sharp jump fluctuation of the market,which is simulated by the local interaction of the oriented percolation system.Moreover,the critical phenomenon of oriented percolation system is used to describe the herding behavior among the participants in the stock market.They simulate the volatility caused by the diffusion of investors' trading attitude and the sudden random jump fluctuation.Furthermore,we will study the finite dimensional distribution of the discrete-time financial price model theoretically.Based on the theory of phase interfaces of stochastic interacting particle system,it is proved that the finite dimensional distribution of the normalized return process converges to the finite dimensional distribution of an Lévy type process.In order to verify the rationality of the model,a series of analysis methods are used to study the volatility properties of the model simulation data set and the real market data set,such as power-law distribution,Lempel-Ziv complexity and fractional sample entropy.The empirical results show that the nonlinear financial model can reproduce some statistical characteristics of real market returns.In Chapter 4,a new microscopic complex jump dynamic model is constructed by using stochastic contact process and compound Poisson process.The stochastic contact process is a continuous time Markov process,which is often regarded as one of the basic models for the spread of certain infectious diseases.The contact process is used to simulate the price fluctuation caused by the mutual influence of investors' trading attitude,and the compound Poisson process is used to describe the stochastic jump volatility caused by macroeconomic environment.In addition,in order to verify the rationality of the financial price model,the finite dimensional distribution of the normalized return process is proved to converge to the finite dimensional distribution of an Lévy type process.In order to better understand the complexity behavior of price dynamic fluctuation,some entropy analyses of the simulation data sets with different parameter sets are carried out,including permutation entropy,fractional permutation entropy,sample entropy and fractional sample entropy.The similar analysis methods are used to analyze the real financial market data set.In Chapter 5,we summarize the main conclusions of this paper. |
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