Asymptotic Normality Of Density Function Of Diffusion Process

Diffusion process is often used to describe some important variables in the financial sector,such as option futures,financial derivatives pricing,oil prices,exchange rates,etc.Until now,the mainstream view believes that random fluctuation is the most fundamental characteristic of the stock market,so modern economists often use the diffusion process to describe the price trend of stocks,so the study on the diffusion process is very meaningful.According to the existing literature,people mainly study the drift function and the estimation of the diffusion function and their corresponding asymptotic properties,while there are few studies on the estimation of the density function of the diffusion process,and there is no discussion on the asymptotic normal convergence rate of this kind of estimation.Therefore,this paper chooses to study the non-parametric estimation of the density function of the diffusion process,specifically studying the density of the first order diffusion process The nonparametric kernel estimator of the function and the nonparametric kernel estimator of the integral diffusion process density function.Under the condition that the diffusion process is smooth and mixed with p,the uniform asymptotic normal convergence rate of the nonparametric estimation of the density function is proved by using the moment inequality,the velocity merging lemma and the method of "large and small blocks".Under some suitable conditions,it is proved that the estimator has a better asymptotic normal convergence rate.In order to prove the theorem,we give some inequalities of mixing process with variable sampling interval in Chapter 5.These inequalities play a key role in the proof of the theorem,and they are also some important research tools in the study of limit theory of mixing process with variable sampling interval.By selecting the closing price data of Moutai stock for empirical analysis,the results show that non-parametric kernel estimation can capture the variation characteristics of density function well.