Uniformly Asymptotic Normality Of Estimation Of The Drift Function For Diffusion Processes

The diffusion model plays a very important role in the financial field.The continuous-time diffusion processes are commonly used to model stochastic dynamic changes in different fields,such as asset pricing,term structure of interest rates and interest rate swap.All the relevant prop-erties of the model can be described and analyzed by the specific properties of these two functions.As a result,the estimation of drift and diffusion functions has become a popular research topic in modern financial statistics.Regarding the nonparametric estimation of the diffusion model,Bandi and Phillips（2003）dis-cussed nonparametric kernel estimations of drift and diffusion functions in diffusion processes and proved the consistency and asymptotic normality of estimators.Later,Nicolau（2007）extended the problem to the integrated diffusion process,constructed nonparametric kernel estimators of drift function and diffusion function based on the infinitesimal moment condition,and proved the weak consistency and asymptotic normality of the estimators.The majority of these studies have been discussing the consistency and asymptotic normality of nonparametric estimation of drift and diffu-sion functions,but none of them have delved into the convergence rate of the asymptotic normality of these two functions.So this paper attempts to study the uniform asymptotic normal convergence rate of the nonparametric kernel estimation of the drift function of the diffusion process and the integrated diffusion process.In this paper,we study the asymptotically normal convergence rate of nonparametric estima-tion of the drift function,under the condition that the diffusion process is-mixing.Using the inequalities of mixing sequences with variable sampling interval given by this paper,we prove the convergence rate of the uniform asymptotic normality of the drift function estimator employing the method of large and small blocks.In the case of optimal bandwidth,the rate of uniformly asymptotic normality reaches n^-2/15.Finally,this paper uses simulated data and the daily closing price data of Shanghai Stock Exchange Composite Index from February 16,2005 to April 1,2022as sample data to investigate the performance of the drift function estimator.The results show that the overall estimation effect of the estimator is good and presents asymptotic normality In addition,some inequalities of variable sampling interval mixing process are given in the second chapter of this paper.These inequalities play a key role in the proof of the theorems,and they are also important tools in researches concerning the limit theory of mixing processes with variable sampling interval.