Asymptotic Normality Of Nonparametric Estimator Of Diffusion Functions

Diffusion process is an important tool to study stochastic phenomena.Especially the diffu-sion process represented by stochastic differential equation,it was widely used in the modeling of stochastic phenomena in physics,chemistry,medicine and finance.In particular,diffusion pro-cesses play a core role in the field of mathematical finance,was widely used to describe some economic variables,such as stock prices,pricing of options and other derivative securities,returns,risk-free interest rates,etc.To study the diffusion process,we need to estimate the diffusion co-efficient and drift coefficient,so the unknown coefficients estimation of this model has become a research hot spot in the field of modern financial statistics.The nonparametric method to estimate the coefficients of the model does not need to set the function form in advance,and can reduce the possible deviation,The nonparametric method for diffusion process has a very broad application prospect in the financial field.In the past,research on nonparametric estimation of diffusion models,Bandi and Phillips(2003)discussed the nonparametric estimator of the drift function and diffusion function of the diffusion process,and proofed the consistency and asymptotic normality of the estimators.Nicolau(2007)extended these works to second-order diffusion processes and proved the consistency and asymptotic normality of nonparametric kernel estimators under appropriate conditions,but neither gave the convergence rate of uniform asymptotic normality.In this paper,under the condition that the diffusion process is stationary and-mixing,the Berry-Esseen type uniform bounds of the nonparametric kernel estimator of the diffusion function are derived by using Bernstein’s large and small block method and the inequality of the mixing sequence with variable sampling interval.By properly selecting the bandwidth and variable sampling interval,we can obtain a good convergence rate of uniformly asymptotical normality.Under the right conditions,by the convergence rate of asymptotically normality of the nonparametric estimator(?)_n~2(x),and combined-rate-theorem.We derived the asymptotic normal convergence rate of the nonparametric estimator(?)_n~2(x)for the second-order diffusion process.It is worth noting that the inequalities for high-frequency mixing sequences given in this paper play a key role in the proofs of our theorems,and they are also important research tools in the study of the limit theory of stochastic processes.Finally,this paper selects the daily closing price of CSI 300 index as the sample data for empirical the analysis.We first analyze the statistical characteristics of the stock index price,then establish a diffusion model for the stock index price.The nonparametric kernel estimator in this paper is used to estimate the diffusion function of the model,then draw the stock index diffusion function chart,which shows”volatility smile”,indicating that higher the absolute values of log-price increments correspond to higher volatility.