复合Poisson单的几点分析-应用数学专业论文.docxVIP

I I 摘 要 复合 Poisson 单是一种特殊的两参数独立增量过程, 也是最典型的状态离散 的两参数马氏过程. 它将在物理学、工程技术学、经济学、金融学等众多领域中有 着十分重要的应用. 许多学者曾对 Poisson 分布、Poisson 过程、复合 Poisson 过 程、Poisson 单和广义 Poisson 单做了大量的工作. 但未见有文献研究复合 Poisson 单. 本文定义了复合 Poisson 单, 讨论了它的基本性质和两参数马氏性, 给出了 它的等价定义, 得到了一系列重要结果, 分析了复合 Poisson 单不存在三点转移 函数族的原因, 利用扩大状态空间的方法, 得到了扩状复合 Poisson 单及其性质 定理, 研究了复合 Poisson 单的可加性, 列举了复合 Poisson 单的应用实例. 本文分四章. 第一章介绍了应用背景、研究现状; 概述本文研究的内容和所 得到的主要结果. 第二章给出了记号、基本概念和以后要用到的结果. 第三章给 出本文研究结果的详细证明，是本文的主要部分. 第四章总结并指出可以继续研 究的问题. 关键词: Poisson 单; 复合 Poisson 单; 独立增量; 矩形增量; Levy 单. I II ABSTRACT The compound Poisson sheet is a special two-parameter independent incre- mental process, and it is also one of the most typical discrete two-parameter Markov processes. It plays an important role in many fields, such as in physics, engineering, economics, financial and so on. Many scholars have done a lot of work on the Poisson distribution, the Poisson process, the compound Poisson process, the Poisson sheet and the generalized Poisson sheet. However, there is no documents on the study of the compound Poisson sheet. In this paper, we define the compound Poisson sheet, discuss the basic prop- erties and the Markov properties of the compound Poisson sheet, point out the equivalent definitions of the compound Poisson sheet, get a series of important re- sults, give the reasons why there is no three-point transfer function family, obtain the definition, the properties and the theories of the extended compound Pois- son sheet by expanding the state space, study the sum of the compound Poisson sheet, and quote the application examples of the compound Poisson sheet. This thesis is divided into four chapters. In chapter one, we introduce the application background and present conditions of research, and generalize the content of this thesis and the main results. In chapter two, notations, definitions and needed results are quoted. In chapter three, we give detailed proofs for the results of the research. It is the main part of this pap