The Application Of Minimum Hellinger Distance Method In Parametric Estimation Of Diffusion Processes

Diffusion processes play an important role in the field of mathematical finance,such as the theory of term structure of interest rate and portfolio selection and asset pricing,derivative pricing and so on,diffusion process is used in all these areas.Then the use of diffusion process can confirm the diffusion process is one of the most attractive tool to describe the financial markets.With the development of modern mathematical finance and the gradual application of diffusion process in the real financial markets,the statistical analysis of diffusion processes has become a hot and important issue in financial mathematics.However,in most of the parametric estimation,they are based on the invariant distribution density of diffusion process.Because the density of invariant distribution of diffusion process does not well reflect the dynamic characteristics of the process within a limited period of time,and in many cases,most of the diffusion process do not have the density of invariant distribution.But the transition density of diffusion process is more important to model testing.And because of the Markov property of the diffusion process,most of diffusion process have the unchanged transition density,and its transition density can get all the dynamic features of the continuous-time process,so the estimation based on the transition density is more meaningful to the estimation based on the invariant density.Therefore,in order to study the parametric estimation of the diffusion process,in this paper,based on the minimum Hellinger distance,a parameter estimator based on transition density is established.First,this paper gives the nonparametric estimator of the density of the invariant distribution and joint density,then it gives the property of them(consistency and asymptotic normality).For the above mentioned,this paper gives the nonparametric estimator of transition density and gives the studies of its properties(consistency and asymptotic normality).Then,the parametric estimator of the diffusion process is structured based on the minimization of the Hellinger distance between the transition density of the diffusion process and a nonparametric estimator of the density.Likewise,the properties of the parametric estimator give in the paper(consistency and asymptotic normality).Finally,to illustrate the properties of the estimator of the parameters,this paper applies the method to two examples:Geometric Brownian Motion and CEV model.