A Time Dependent Diffusion Equation Of Statistical Inference

The subject of statistical inference for stochastic differential equation has been and isstill of considerable interest due to its applications in finance. Nonparametric estimation ofthe 1-dimensional diffusion coefficient has been well studied in Financial Mathematics.But in finance, the economic conditions change from time to time. Thus, it is reasonable toexpect that the instantaneous expected return and volatility depend on both time and pricelevel for a given state variable, then there have been increasing needs on the study oftime-dependent diffusion models. Various efforts have been made to explicitly express thedependence of parameters on time. For these reasons, it is necessary to study thenonparametric estimation problem in time dependent diffusion models.In this paper, we first reviewed the fruit of diffusion estimation in theory andapplications., then discussed the nonparametric estimation problem in time dependentdiffusion models. The thesis is divided into two parts:In chapter 2 we discussed the estimation of the diffusion efficient. First we translatethe estimation problem into the nonparametric regression by Girsanov theorem. Thenthrough dividing the time span into many blocks, the 2-dimentional estimation problembecame the 1-dimentional problem. We applied the local polynomial estimation on it andachieved the estimator of diffusion efficient. Further more we proved the asymptoticnormality and consistency of it.In chapter 3, we tested our techniques by simulating three data sets from specificmodels and obtained the good results and then applied it to the Shanghai A Share Index. Inthe last, we proposed the method to examine the model time-independent or not and testedit by simulations.