Investigation On Financial Market Fluctuations Of Stochastic Interacting Particle Systems And Volatility Statistics

Price fluctuations in the financial market have always been one of the research focuses in the field of financial mathematics and financial engineering.The exploration in this field can help scholars and market participants to more deeply understand evolution mechanisms and behavioral characteristics of financial market dynamics,explore internal laws of the financial market,which is of great significance in financial risk management,asset valuation,derivative pricing,portfolio management and so on.In recent years,with the continuous application of mathematics,information science,physics,systems science and other related sciences in the financial field,a large number of interdisciplinary subjects have gradually formed.Among them,the emergence of “Econophysics” as an important subject provides a new perspective and more abundant research methods for the study of price fluctuation behavior.Based on the related ideas and theories of “Econophysics”,this paper mainly conducts innovative exploration and research on the price fluctuation behavior of the financial market from two aspects:establishing price fluctuation model and constructing volatility statistics.The main innovations of this paper are as follows.On the one hand,this paper introduces three important stochastic interacting particle systems,including finite range stochastic interacting epidemic system,lattice voter model and voter model on small-world network,in the study of price fluctuation models in the financial market,which can describe the dissemination mechanism of information and the interaction behavior of investors in the market from a new perspective.Then combined with the theoretical knowledge of stochastic process and random jump process,three new stochastic interacting price fluctuation models are constructed to discuss and explain the formation mechanism and law of financial market price fluctuations.In the validation of the rationality of the models,this paper selects powerful research methods to analyze and compare the statistical characteristics and nonlinear properties of the simulated data from the three new price fluctuation models under the multi-dimensional parameters and the real financial market data,and the similarity of the results verifies the rationality of the models.By simulating the investment strategies and interacting behaviors of market investors,this work explores the formation mechanism and evolvement law of price fluctuations in the financial market,and provides the theoretical basis and micro dynamic models for the study of price fluctuations.At the same time,some new nonlinear methods are proposed,which provide new tools for analyzing the properties of price fluctuations in the financial market,including the index fluctuation fuzzy entropy and the composite distance fuzzy entropy.On the other hand,through study of the intrinsic structural characteristics of the price fluctuation series in the financial market,this paper constructs two new volatility statistics to describe price volatility dynamics from the volatility shortest passage time and the volatility change intensity.This work is an innovation and exploration of volatility research,depicting the changing trend and risks of price fluctuations in financial markets from a new perspective,and enhancing the knowledge and understanding of market participants on the law of price fluctuations.Main works of this paper are as follows.Chapter 1 introduces the research background,theoretical basis,innovations and main research contents of this paper.Chapter 2 firstly introduces a new stochastic interacting epidemic price fluctuation model based on the finite range stochastic interacting epidemic system.As an important stochastic interacting particle system,the finite range stochastic interacting epidemic system can describe the complex dynamic mechanism of the transmission of a disease among individuals in space.The states of individuals can be divided into three categories: healthy,infected and immune,and will change according to certain rules under the influence of the disease.The stochastic interacting epidemic price fluctuation model uses individuals in the system to simulate investors in the financial market,and at the same time uses the transmission mechanism of this disease among individuals to simulate the interaction mechanism among investors under the influence of market information,in which the spread and change of investors' investment attitudes are considered to cause price fluctuations of financial products.The disease in the finite range stochastic interacting epidemic system has a wider spread range,so each investor in the model can exchange information with more investors.After establishing the model,this chapter discusses some statistical and nonlinear behaviors of simulated data under different parameter combinations.In this process,the same methods are used to study and analyze the data sets of the real market,and the validity of stochastic interacting epidemic price fluctuation model is verified by comparing the above fluctuation characteristics.The new index fluctuation fuzzy entropy method is proposed for the first time to study the complexity behavior of financial time series from multiple dimensions through four metrics.Chapter 3 combines the random jump process with lattice voter model and voter model on small-world network respectively for the first time,and establishes the stochastic interacting lattice voter price fluctuation model and the stochastic interacting small-world network voter price fluctuation model.The innovation of the two models lies in the description of the formation mechanism of market price fluctuation from the internal micro-effects and the external macro-environment.On the one hand,the voter theory is introduced into the lattice space and small-world network structure respectively,to simulate the frequent and normal price fluctuations caused by the mutual influence among different investors and changes in investors' investment attitudes.On the other hand,a random jump process is constructed by using compound Poisson process,which is used to capture the sudden and violent price fluctuations caused by external macro events in the financial market,and explain the random sudden jumps in the price evolution process.At the same time,this chapter makes a series of analyses and comparisons between the simulated data of the model under different parameter combinations and the data of the real market to explore fluctuation behaviors and complex natures,which can verify the validity of the stochastic interacting lattice voter price fluctuation model and the stochastic interacting small-world network voter price fluctuation model.In the process of measuring the complexity of financial time series,the composite distance fuzzy entropy method introduced in this chapter shows higher accuracy and stronger robustness than the fuzzy entropy method,which can provide important help for the study of the complexity of price fluctuations.Chapter 4 introduces two new volatility statistics,including volatility two-component range intensity and cumulative volatility two-component range intensity.Among them,volatility two-component range intensity considers the duration of the local price volatility and the volatility maximum change intensity,while cumulative volatility two-component range intensity uses the volatility cumulative change intensity to replace the volatility maximum change intensity in volatility two-component range intensity.These two volatility statistics establish the relationship between duration and range of change in the price volatility series.They can help investors to grasp the change trend and length information of the current fluctuations,and reduce the impact of some extreme fluctuations on market participants,which can provide reference for market participants in risk measurement.This chapter studies the basic statistical characteristics,probability distribution,power-law scaling of volatility two-component range intensity series and cumulative volatility two-component range intensity series of the real financial products in the financial market.And the autocorrelation and correlation of the above time series are discussed by nonlinear methods in this paper.The above results enrich the research on price fluctuations in the financial market.Chapters 5 introduces the main conclusions of this paper.