Mathematical Sciences: Stochastic Processes

数学科学：随机过程

• 批准号: 9503519
• 负责人: Michael Marcus
• 金额: $ 21万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-15 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9503519&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Stochastic; Processes

9503519 Marcus Abstract The investigators will continue their study of continuity properties of additive functionals of Markov processes and their limiting behavior. Using an isomorphism theorem of Dynkin and results from the theory of probability in Banach spaces and random Fourier series they will explore relationships between continuous additive functionals of Markov processes and Gaussian processes and chaoses and random Fourier series. In this way they expect to be able to use known results about each of the different processes to obtain new results about the other ones as they did in their work on local times and Gaussian processes. They also plan to use techniques developed in their earlier work to study the number of intersections of random walks, particularly in the critical and super-critical cases. Motivated by the importance of a special class of second order Gaussian chaoses related tm Wick squares in the study of positive continuous additive functionals, they will try to find necessary and sufficient conditions for the continuity of these chaoses. They will also attempt to use self-collision local times of superprocesses to construct a self-interacting measure valued diffusion and study the large deviations of additive functionals of super Brownian motion. This research deals with fundamental properties of stochastic processes and has potential application in all areas that deal with random phenomena. The investigators will consider data that evolves in time in a random fashion, such as electrical signals received from distant satellites or dollar values in financial transactions. In particular they will consider the amount of time the data takes specific values or is in a small, heavily weighted, range of values and how this amount of time varies as the set of specific values vary. They plan to do this for a wide class of processes called Levy processes. These processes have often been thought of as models for many important practical applications but, so far, they have b een considered too difficult to analyze very deeply. The authors plan to use several new and powerful methods, recently developed by themselves and others, to accomplish this deep analysis.

9503519马库斯摘要 研究人员将继续研究马尔可夫过程的可加泛函的连续性及其极限行为。利用同构定理Dynkin和结果从理论的概率在Banach空间和随机傅立叶级数，他们将探讨连续添加剂泛函的马尔可夫过程和高斯过程和混沌和随机傅立叶级数之间的关系。通过这种方式，他们希望能够使用关于每个不同过程的已知结果来获得关于其他过程的新结果，就像他们在当地时间和高斯过程的工作中所做的那样。他们还计划使用他们早期工作中开发的技术来研究随机行走的交叉点数量，特别是在临界和超临界情况下。受一类特殊的二阶高斯混沌在正连续可加泛函研究中的重要性的启发，他们将试图找到这些混沌连续的充分必要条件。他们还将尝试利用超过程的自碰撞局部时来构造自相互作用测度值扩散，并研究超布朗运动的可加泛函的大偏差。 这项研究涉及随机过程的基本性质，并在处理随机现象的所有领域都有潜在的应用。调查人员将考虑以随机方式随时间演变的数据，例如从远处卫星接收的电信号或金融交易中的美元价值。特别是，他们将考虑数据采用特定值或处于小的、重加权的值范围内的时间量，以及该时间量如何随着特定值集的变化而变化。他们计划对一类广泛的过程进行这种处理，称为Levy过程。这些过程通常被认为是许多重要的实际应用的模型，但到目前为止，它们被认为太难了，无法进行非常深入的分析。作者计划使用几种新的和强大的方法，最近开发的自己和他人，完成这一深入的分析。