Parameter Estimation And Application Of Geometric Mixed Fraction Brownian Motion Model

China’s stock market development has been nearly 30 years of history,but compared with the western world and the United States and other advanced economies of the stock market,China’s stock capital market still has more imperfections,so scientific judgment of the future trend of China’s stock market development is an urgent problem to be solved.The current stock forecasting methods can be roughly divided into three categories.One is the statistical time series method,which takes into account the time series characteristics of financial data and uses the time series models such as the ARM A model,ARIMA model and GARCH model to continuously innovate and reform.The model has a good effect on short-term market forecasts,but it has poor effect on long-term market forecasts,and it cannot effectively deal with the complex nonlinear relationships in the stock market.The second category is to consider the huge amount of financial data,with a variety of data sample sizes,and consider applying the computer or applied intelligent computing in the big data calculation method to the financial field,such as machine learning,deep learning,artificial intelligence,and training models to achieve certain accuracy through algorithms Requirement,but easy to cause local optimum.The third category is mathematical financial methods,which use random particle characteristics to describe the fluctuations of financial asset prices through stochastic processes and no-arbitrage theory.Mathematical theory and methods continue to deepen in financial economics and have become a leader in the financial cross academic field,and have achieved many results.This is also the main research model of this article.The geometric Brownian motion model in mathematical finance is a typicalfinancial stochastic model used to describe the evolution of stock prices over time in modern mathematical finance,and has a very wide range of applications.The geometric Brownian motion model can not only accurately describe the process of stock price evolution over time,but also reveal the movement law of stock price fluctuations which contain a linear trend.It uses historical data to estimate parameters and use these parameters to estimate future financial indicators.After long-term empirical research on the randomness described by standard Brownian motion,it is found that the time series of stock prices is not only a biased random walk,but also has characteristics such as long-term dependence.Then the concept of fractional Brownian motion is introduced,and the concept of fractional Brownian motion is proposed.It has profound practical and theoretical significance in the mathematics-finance field.Fractional Brownian motion is used instead of Brownian motion and stochastic differential equations driven by it to describe financial time series data.Subsequently,the mixed fraction Brownian motion was included in the research scope.As for the geometric Brownian motion driven by the mixed fraction Brownian motion,people have not carried out related explorations.The geometric Brownian motion driven by the mixed fraction Brownian motion is more important than the geometric Brownian motion,the fractional Brownian motion and the mixed fractional Brownian motion.Perfect,it can be supplemented on the basis of the original research,promote the application of geometric Brownian motion in option pricing and other financial derivatives,and can better characterize the stock price.Therefore,the research on geometric Brownian motion driven by mixed fraction Brownian motion It has more theoretical and practical significance.The main research work of this paper is to construct the geometric mixed fraction Brownian motion model and the parameter estimation in the model,and use real screening data for empirical application.First,use the linear combination of the standard Brownian motion and the fractional Brownian motion to construct the Brownian motion term,and use the re-scalar range method(R/S method)to estimate the Hurst exponent of the fractional Brownian motion;secondly,use two kinds of parameters for the model Method to estimate the parameters of the fluctuation coefficient and drift coefficient.One method is to use the maximum likelihood method to estimate and use the method of solving the binary equations in the numerical simulation to obtain the estimated value.The other method is based on the quadratic variation method.After the volatility coefficient is estimated,the maximum likelihood estimation method is used to give the estimator of the drift coefficient,and the derivation proves the asymptotic nature of the estimator;then the Monte Carlo simulation is used to investigate the mean value of the estimated value and the estimator of variance analysis Accuracy;finally in the empirical part,the CSI 300 stock data is screened to construct an effective data set with Hurst index greater than 0.5,and the parameter estimates obtained by the maximum likelihood estimation under the quadratic variation numerical method are given,And use the estimated parameter values to generate the geometric mixture fraction Brownian motion orbit,and compare the theoretical orbit diagram with the actual orbit diagram.The results show that geometric Brownian motion driven by mixed fraction Brownian motion is more complete and can better predict stock prices,and can also describe financial and economic phenomena.Compared with geometric Brownian motion model,geometric fraction Brownian motion model,mixed Fractional Brownian motion model is more general,more in line with practical problems,and can better describe the changes in stock prices,which can be better applied to options pricing in the future.For the parameter estimation problem in the model,it is found through experiments that it is better to use the quadratic variation method and the maximum likelihood estimation method in practice.The parameter estimation separately not only makes it more practical,but also can effectively improve the efficiency.