Nonparametric Estimation Of Drift And Diffusion Functions In Diffusion Process

Diffusion process plays an important role in the financial market, which is widely used to describe a number of economic variables, such as stock prices, options and other derivative securities pricing, returns, interest rate data etc.This paper mainly studies the estimation of drift and diffusion functions in homogeneous diffusion process and its applications. In the estimation of diffu-sion, we give a new time domain estimation method, and combined it with state domain estimation methods, such as kernel function estimation method and local polynomial estimation method. Then we use our method and some other meth-ods in treasury bill data and simulation data. Specifically, the main work of this article includes the following three parts:Part I:We give a nonparametric estimation for the diffusion functionσ2(.) of a homogeneous diffusion process, we mainly consider the time domain and state domain dynamic combination of estimation methods in the estimation for the diffusion functionσ2(.) of a homogeneous diffusion process. In time domain, we propose a new power smoothing method And illustrate the power smoothing's two advantages. Then we give a dynamic combination between power smoothing method and kernel function estimation, or a dynamic combination between power smoothing method and local polynomial estimation method.Part II:We give a nonparametric estimation for the drift functionμ(.) of a homogeneous diffusion process. We can get the nonparametric estimation of diffusion function from Part I, then use the relationship between drift function, diffusion function and the stationary density of diffusion process We can get the estimation of the drift function. We use kernel function estimation in estimating the stationary density. PartⅢ:In partⅠ, we have given a new dynamic combination method between time-domain and state-domain, we use our method in treasury bill data and simulation data. We consider many estimation method in time domain,state domain,combination of time domain and state domain, specifically;In time domain, we consider power smoothing method (T), exponential smoothing method (TF), Epanechnikov kernel function method (TY).In state domain, we consider kernel function estimation method(K), local polynomial estimation method(L), combination method between kernel function estimation method and local polynomial estimation method(K-L).In combination of time domain and state domain, we consider a dynamic combination between power smoothing method and kernel function estimation(T-K), a dynamic combination between power smoothing method and local polyno-mial estimation method(T-L),a dynamic combination between exponential smooth-ing method and local polynomial estimation method(T-L(Fan)), a dynamic com-bination between Epanechnikov kernel function method and local polynomial es-timation method(T-L(Ye)), besides, we also consider non-parametric Bayesian estimation method.By the above method, we can get the diffusion estimation of diffusion process ,then applied MADE, RADE, MSE, IMADE and RIMADE as evaluation criterion of non-parametric estimation, by comparing various methods, we find our method has advantage compared to other methods, no matter using treasury bill data or simulation data.