Model Selection Based On Levy Process Function

This article use the method of penalty function to choose a model of Levy process in the space of L2. As the Khinchine decomposition theorem, we known the Levy process is composed of triples (A,a,v). So just make sure the triples will be able to determine the Levy process. A and a mainly determine the Brownian movement which with drift, and the measure v determines the behavior of the pure jump of the Levy process.On the linear space{Sm,m∈M},we will give the model of Gaussian process and Compound Poisson process. By the definition of the linear Gaussian model, we know as long as the mean function of Gaussian process is determined, the model of Gaussian process will be obtained. So we use the method of penalty function to do the model selection of mean function of the Gaussian process in the space L2.By Wiener-Khinchine decomposition theorem, we know Compound Poisson process is decided by Levy measure v. Assume that the measure v absolutely continuous with a known measure η in space L2, then by Radon-Nikodym derivative dv(x)/dη(x)=s(x),x∈D, regularization density function s(x) regarding the measure η. And through the model of density function s(x) is given in the space of L2, the Compound Poisson process model will be decided. Through the determination of respectively for the two parts model, we obtained the model of Levy process.In conclusion, the model selection method is mainly in two steps. The first step Estimators of the linear model is given by the least squares estimate. Step Two:We select the optimal estimator among the least squares estimators by the way of penalty function, and the estimator should satisfies the Oracle inequality in the penalty function with certain conditions. The full text mainly includes the following four parts.The chapter one introduces the background and practical significance of the problem of this paper. Reviews the research achievements of predecessors, then puts forward the issue to be studied in this paper. Research methods and main conclusions are presented.The chapter two gives the knowledge of Levy process, Gaussian process and Compound Poisson process.The chapter three gives the model selection method of Gaussian process in space L2. Prove there is a punishment s to satisfy projected Oracle inequality estimates. when the penalty function have certain conditions.The chapter four, in space L2, we will give the model selection of Poisson process. Then prove exist a punishment projection estimator s which meet the Oracle inequality, when the penalty function satisfies certain conditions. And through v(dx)=s(x)ηd(x), we can have v to determine the model of Compound Poisson process.