The Parameter Estimation Problem For Several Classes Of Stochastic Differential Equations

In recent decades,stochastic differential equation is applied widely in many re-search fields.In this thesis,we mainly consider the parameter estimation of two order nonlinear stochastic differential equations with small perturbations.The parameters of a are estimated by using the maximum likelihood estimation method.The properties of the estimator are discussed as the small parameter goes to zero or time T goes to in-finity.Typically,the unbiasedness property and consistency of the estimator as ??0 are proved.Then in the case of fixed ? and ??0,the asymptotic distributions of the estimator are given as T goes to infinity respectively.Last the numerical simulation is given to show the unbiasedness property and asymptotic normality of estimator.This thesis is divided into five parts.The first chapter is the introduction,includ-ing research background,research significance and so on.The second chapter is the preliminary knowledge,including the basic concepts involved in this thesis,and reviews the some basic knowledge of stochastic analysis.In the third chapter,a class of non-linear stochastic differential equations is introduced,and the parameter ? is estimated,and the effects of the time T on the properties of the statistics are discussed.In the fourth chapter,the parameter estimation of stochastic differential equations with small perturbations is discussed,and the parameter is estimated by the maximum likelihood estimation method,and the effects of the small parameter ? and time T on the statis-tical characteristics of the estimator are studied.Finally,the numerical simulation of MATLAB is carried out to obtain the conclusion that the smaller the parameter ? is,the more the ?'s estimate is close to the true value.In the last chapter,an improved and more complicated model is introduced,which includes the effects both standard Brownian motion B_t and another independent Brownian motion Wt.On the basis of the above,we discuss the unbiasedness property and asymptotic distribution of the parameter estimators.At the end of this chapter,we use the numerical simulation to deepen the understanding of the nature of statistics.