Parameter Estimation Of Two Classes Of Stochastic Financial Models Driven By Fractional Lévy Processes

Random phenomena exist in every field,stochastic differential equations are important tools to describe random phenomena,the corresponding research theories are widely used in finance,medicine,aerospace and other fields.Stochastic differential equations often contain unknown parameters,so the estimation of unknown parameters is an important research direction in the field of stochastic analysis.Although some of the parameters of stochastic economic models have been solved,most of them are driven by Brownian motion.There are few studies on establishing models driven by fractional Levy processes and performing statistical inferences,and most of the equations in the existing literatures only contain an unknown parameter.This paper focuses on parameter estimation and statistical inference of stochastic financial models driven by fractional Levy processes.The model contains two unknown parameters.The consistency of the estimator and the asymptotic distribution of the estimation error and other properties has been studied in the paper.verifying the validity of the estimator and the estimation method shows that the use of this model can more accurately grasp the dynamic changes of assets,and the research has certain practical value and innovative significance.The full text is divided into four parts.The first chapter introduces the general situation of the development of the diffusion process,expounds the research dynamics and practical value of the parameter estimation of two types of stochastic financial models driven by the fractional Levy process,the research questions and research significance of this paper;The second chapter conducts parameter estimation and hypothesis testing for the Vasicek model based on discrete observations.The main steps are firstly to introduce the contrast function,and obtain the least squares estimator according to the relationship between the extremum and the partial derivative,and then according to the Markov inequality,the Gronwall inequality,and the Cauchy-Schwarz The consistency and asymptotic distribution of the estimator are obtained from the inequality,and finally the least square estimator of the coefficient of the drift term of the stochastic differential equation converges to the true value of the parameter according to the probability,and the validity of the estimation method is verified by numerical simulation.The third chapter uses similar steps to estimate the parameters of the CIR model under discrete observations.The CIR model is an extension of the Vasicek model,which solves the problem of possible negative interest rates.Similar to the steps in Chapter 2,the parameter estimators and their specific analytical expressions are calculated first.Secondly,the strong consistency of the parameter estimators of the CIR model and the asymptotic normality of the estimation errors have been proved.Numerical experiment results show that when n is large enough and small enough,the estimator obtained is very close to the true value of the parameter.The fourth chapter is the summary and prospect.It summarizes the content of the full text,briefly expounds the solved problems and unsolved problems,and puts forward some potential research directions.