Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của trường đại học vinh năm 2009 tác giả: 2. Nguyễn Thanh Diệu, Biểu diễn Doob - Mayer đối với Martingale trên thang thời gian. | DOOB - MEYER DECOPOSITION FOR SUBMARTINGALES ON TIME SCALES NGUYEN THANH DIEU a Abstract. The aim of this paper is to study the Doob - Meyer decomposition for a submartingale on time scales. The obtained results can be considered as a generalization of the Doob - Meyer decomposition for submartingale in the discrete and continuous time. Introduction The Doob - Meyer decomposition theorem for a submartingale is one of the central topic in the probability theory. In 8 . Meyer has proved that a submartingale belonging to the class D admits a unique decomposition into a sum of a uniformly integrable martingale and a predictable integrable increasing process. Later on this result is considered in the continuous time in 10 by using the increasing natural process instead the concept of prediction. Moreover in recent years the theory of dynamic on time scales which was introduced by Stefan Hilger in his PhD thesis 5 has been born in order to unify continuous and discrete analysis. Since then this problem has received much attention from many research groups. Therefore it is natural that we need to transfer this theory to the so-called stochastic calculus on the time scale. The first attempt of this topic is to consider the Doob - Meyer decomposition for submartingales indexed by a time scale and this is the aim of this paper. The obtained result can be considered as a common method to present Doob - Meyer decomposition in the discrete and continuous time. The organization of this paper is as follows. In section 1 we survey some basic notation and properties of the analysis on time scale. In section 2 we presents the main result of Doob - Meyer decomposition theorem for a submartingale on time scale. 1. Preliminaries on time scales This section surveys some notations on the theory of the analysis on time scales which was introduced by Stefan Hilger 1988 5 A time scale is a nonempty closed subset of the real numbers R and we usually denote it by the symbol T. We assume .
