Simulation And Application Of A Kind Of Long Memory Stochastic Process

Since the 20 th century,Brownian motion and the theory of stochastic differential equations have gradually improved,which is often used to describe random phenomena such as price fluctuations and signal changes,and have more and more extensive applications in financial markets,physical engineering,and biomedical fields.With a deeper understanding of random phenomena,long-memory features are more often shown in data such as return on assets and network communications,which is not consistent with the statistical characteristics of the Brownian motion.In order to describe the data more accurately,it is the fractional Brownian motion(fBm,in short)with long memory that draws the attention of statisticians.Theoretical research on the long-memory process has been carried out and gradually applied to the fields of geophysics and biomedicine,which has further promoted the development of network communication and financial market theory.In the pricing of financial derivatives,compared to the Ornstein-Uhlenbeck(O-U,in short)process driven by Brownian motion,the fractional O-U process driven by fractional Brownian motion has both mean recovery and long memory,which is more suitable for asset value changes in option pricing and risk management.It becomes the most typical type of random process in the long memory model.In addition,the signal distribution in network communication,typhoon pulsating wind speed in meteorology and logarithmic return on stock prices are all non-Gaussian.At this time,the Rosenblatt process with non-Gaussian properties can better describe this type of process,and it is a long-memory random process like fBm.In this paper,we mainly study the data simulation and application of fractional O-U process driven by fBm and Rosenblatt process.For the former issue,the data simulation and empirical analysis of parameter estimation in the form of spectral density approximation and Donsker approximation of the fractional O-U process are given,and the statistical properties of the estimation and program operation are compared.For the latter problem,the maximum likelihood estimation of parameters is mainly studied,and the simulation results confirm the validity of the estimation.Firstly,the paper introduces the research background and meaning,related research results and the basic theories involved in the article.Secondly,by means of data simulation and empirical analysis,the parameter estimates of fBm-driven fractional O-U process in spectral density and Donsker approximation are compared.On the one hand,the maximum likelihood estimates of the parameters of the fractional O-U process in two approximations are given,and through numerical simulation,the results are compared in terms of statistical properties and program operation.On the other hand,performed with the closing prices of CSI 300,the empirical analysis compares the simulated orbits with the actual orbits.Thirdly,the parameter estimation of the fractional O-U process driven by the Rosenblatt process is studied.On the basis of the optimization model,the maximum likelihood estimator of the fractional O-U process in Donsker approximation is given,and the validity of the estimation is proved by numerical simulation.Finally,we summarize the research content of the paper,point out the limitations of this article,and discuss the future research direction.Based on numerical simulation and empirical results,this paper analyzes and evaluates the parameter estimation of the fractional O-U process under the spectral density approximation and the Donsker approximation.It lays the foundation for the systematic research of fractional O-U process,and also provides a basis for data analysts to make decision.Moreover,this paper studies the maximum likelihood estimation of the fractional O-U process driven by the Rosenblatt process.The optimized fractional O-U process is more general and more suitable for non-Gaussian situations.It is of great significance to improve the long-memory theory and the development of many fields such as financial markets.