Complex Stochastic Systems

复杂随机系统

• 批准号: 0205034
• 负责人: Thomas Kurtz
• 金额: $ 39.24万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-05-15 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0205034&HistoricalAwards=false
• 关键词: Complex; Stochastic; Systems

This research program will focus on singular martingale problems and stochastic differential equations, particle representations for stochastic partial differential equations and measure-valued processes, and spatial point processes. The work on singular martingale problems and related stochastic equations addresses fundamental questions regarding the specification, transformation and approximation of the models. Applications to controlled stochastic networks motivate much of the work. Particle representations provide a powerful tool for understanding the behavior of complex stochastic processes. Work to be performed will extend the range of applicability of these methods, develop new constructions of particle representations, and address problems of existence and uniqueness of solutions of the stochastic equations underlying these representations. Application of the results to filtering and to theoretical models in finance will provide both useful intuition and a check on the appropriateness of the theoretical developments. Spatial point processes characterized as stationary distributions of Markov birth and death processes and as solutions of stochastic equations driven by Poisson random measures will be studied. The work will focus on spatial ergodicity and central limit theorems with applications to parameter estimation in mind. The study of stochastic processes is concerned with mathematical descriptions of natural phenomena governed by "random" or "chance" mechanisms. Mathematical models of such phenomena may attempt to describe variation in time, in space, or both. The research to be performed is concerned with developing methods for specifying these mathematical models, approximating complex models by simpler ones, obtaining information about the true state of a system from corrupted or noisy observations, and determining how to influence or "control" the evolution of the models and the phenomena they represent. Motivating examples include models for the effects on asset prices of the valuations assigned by a large number of traders and for communication and computer networks.

该研究计划将侧重于奇异鞅问题和随机微分方程，随机偏微分方程和测度值过程的粒子表示，以及空间点过程。奇异鞅问题和相关的随机方程的工作地址的规范，转换和近似模型的基本问题。控制随机网络的应用激发了大量的工作。粒子表示为理解复杂随机过程的行为提供了一个强有力的工具。要进行的工作将扩大这些方法的适用范围，开发新的粒子表示的建设，并解决这些表示的随机方程的解的存在性和唯一性的问题。将结果应用于金融中的过滤和理论模型，将提供有用的直觉和对理论发展的适当性的检查。将研究马尔可夫生灭过程的平稳分布和由泊松随机测度驱动的随机方程的解。这项工作将集中在空间遍历性和中心极限定理与应用程序的参数估计铭记。随机过程的研究关注的是由“随机”或“机会”机制控制的自然现象的数学描述。这种现象的数学模型可能试图描述时间、空间或两者的变化。要进行的研究是关注开发方法来指定这些数学模型，近似复杂的模型由简单的，从损坏的或嘈杂的观察系统的真实状态的信息，并确定如何影响或“控制”的演变的模型和它们所代表的现象。激励的例子包括模型的资产价格的影响分配的估值由大量的交易商和通信和计算机网络。