Complex Stochastic Systems

这项研究计划将集中在奇异鞅问题和随机微分方程，随机偏微分方程和测值过程的粒子表示，以及空间点过程。关于奇异鞅问题和相关随机方程的工作解决了关于模型的规范、变换和逼近的基本问题。对受控随机网络的应用推动了大部分工作。粒子表示法为理解复杂随机过程的行为提供了有力的工具。将要进行的工作将扩大这些方法的适用范围，开发新的粒子表示结构，并解决这些表示所依据的随机方程解的存在和唯一性问题。将结果应用于金融中的过滤和理论模型，将提供有用的直观和对理论发展的适当性的检查。以马尔可夫生灭过程的平稳分布和泊松随机测度驱动的随机方程的解为特征的空间点过程被研究。这项工作将集中在空间遍历和中心极限定理上，并将应用到参数估计中。随机过程的研究涉及到对由“随机”或“随机”机制支配的自然现象的数学描述。这类现象的数学模型可能试图描述时间、空间或两者的变化。将要进行的研究涉及开发方法来指定这些数学模型，用更简单的模型来近似复杂的模型，从受污染或有噪声的观测中获得关于系统真实状态的信息，以及确定如何影响或“控制”模型的演变及其所代表的现象。令人振奋的例子包括大量交易员分配的估值对资产价格的影响以及通信和计算机网络的模型。

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.