Variance And Bias Reduction Estimation For First Order Non-stationary Diffusion Process

In the field of modern financial mathematics,the diffusion process plays a central role.It has been well applied in terms of asset pricing,derivatives pric-ing,and the term structure of interest rates.It can be said that it is the most attractive tools to describe the financial market.But in the study of diffusion process,most of the models are assumed to be stationary.However,in practical applications,the assumption of stationarity is very harsh,especially for financial data,which is very difficult to be stationary.Many existing studies and empirical studies have pointed out that common economic variables,such as asset price series,term structure of interest rate,exchange rate,all show non-stationary over time.Therefore,many scholars have focused on the non-stationary diffusion process in recent years.The existing methods for estimating non-stationary diffusion process mainly use symmetric kernel method to estimate diffusion model coefficients non-parametrica-lly.Bandi and Hillis(2003)first systematically studied the nonparametric esti-mators of coefficients of non-stationary diffusion processes.On this basis,Fan and Zhang(2003)constructed the local linear estimator of the coefficients of non-stationary diffusion process to improve it,which effectively reduced the bias at the sparse point.However,due to the limitation of the symmetric kernel method,the estimation effect of the model at the boundary points is notgood,and the boundary effect is obvious.The main work of this paper is to proposes the local linear smoothing to estimate the unknown volatility function in scalar diffusion models based on Gamma asymmetric kernels.The main innovations of this paper are:First,for the first time,local linear estimation and asymmetric kernel estimation are combined to estimate the first-order non-stationary diffusion process.This reduces the variance at the boundary points while retaining the advantage of small deviation of local linear estimation and weakens the boundary effect.Second,for the asymmetric kernel estimator given in this paper,its asymptotic property and its proof process are given.In the process of proving the asymptotic property,the problem that it is difficult to establish asymptotic property because of the predictability of estimator is de-stroyed in the past is solved.The structure of this paper is as follows:In Chapter 1,we will briefly introduce the background knowledge of stochastic analysis and some theorems to be used in the following proofs.In Chapter 2,we will introduce some classical method-s of kernel density estimation.In Chapter 3,we will focus on the background and current situation of the first-order diffusion process.In Chapter 4,we will construct a locally linear asymmetric kernel estimator for the coefficients of first-order non-stationary diffusion processes and prove its asymptotic properties.In Chapter 5,we will test the validity and robustness of our estimators by Monte Carlo simulation.