Nonparametric Estimation Of Second-Order Diffusion Processes

Stock prices and futures options pricing in financial markets can usually be effectively expressed by random diffusion processes.The diffusion coefficient σ（x）is usually used to describe the uncertainty of asset returns,reflecting the risk level.Therefore,the estimation and prediction of the drift coefficient function μ（x）and the diffusion coefficient functionσ（x）of the stochastic diffusion model have become the focus of research in the financial field.Bandi＆Philli（2003）were the first to use local time correlation theory to study the strong consistency and asymptotic normal properties of nonparametric kernel estimates of diffusion processes.Nicolau（2007）generalized these works to second-order diffusion processes,giving weakly consistent and asymptotically normal properties of N-W kernel estimators under appropriate conditions.But he did not prove convergence almost everywhere.In this paper,the density function p（x）,the drift coefficient μ（x）and the diffusion coefficient σ（x）of the second-order diffusion model obtained by the N-W kernel regression estimation method are nonparametric kernel estimators pn（x）,μn（x）and σn2（x）.And prove that these kernel estimators converge almost everywhere.This paper introduces the intermediate quantity pn（x）.Then convert the proof pn（x）-p（x）=oas（1）into a step-by-step proof pn（x）-p（x）=oas（1）and pn（x）-pn（x）=oas（1）.We prove the strong consistency of nonparametric kernel estimators under relatively mild conditions using the important moment inequalities of p-mixed sequences.At the same time,our conclusion simplifies some conditions in Nicolau（2007）.In the numerical simulation part,our results show that the average deviation of the drift coefficient estimator μn（x）is 0.1131,and the diffusion coefficient estimator σn2（x）is 0.0645.It shows that the nonparametric kernel estimation can better capture the variation characteristics of the model drift coefficient μ（x）and diffusion coefficient σ2（x）.However,the analysis results of the estimated-true value comparison chart show that the estimated diffusion coefficient σ2（x）tends to overestimate as a whole.This is caused by the defect of the N-W diffusion coefficient estimator σn2（x）itself.This problem can be solved by adding a correction bias term.