Complex Stochastic Systems

9971571库尔茨研究项目将在随机分析、测量值过程、随机滤波和控制、大偏差和空间点过程等领域开展。在随机分析领域的工作将集中在随机偏微分方程，随机方程的模拟方案的分析，以及随机方程和鞅问题之间的关系。推导并分析了非标准观测结构的滤波方程。以前关于正则控制问题的鞅表征的工作将扩展到奇异和部分观察控制问题。基于非线性半群收敛的马尔可夫过程的大偏差研究方法将扩展到新的模型类别。空间点过程特征为马尔可夫出生和死亡过程的平稳分布和作为泊松随机测度驱动的随机方程的解将被研究。工作将集中于空间遍历性和中心极限定理，并考虑到参数估计的应用。随机过程的研究涉及对受“随机”或“偶然”机制支配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。要进行的研究涉及开发方法来指定这些数学模型，用更简单的模型近似复杂的模型，从损坏或“嘈杂”的观测中获得关于系统真实状态的信息，并确定如何影响或“控制”模型及其所代表的现象的演变。特别是，过滤方程提供了一种处理关于不能直接和/或准确观察到的现象的信息的方法，通过使用未观察到的现象和可以观察到的量之间的关系的数学模型。将考虑的情况类型的示例包括观测信息在空间上分布的模型，例如可能从卫星观测中获得的模型，以及试图从网络中几个点的观测中获得有关网络状态的信息的通信网络模型。

9971571Kurtz Research programs will be carried out in the areas of stochastic analysis, measure-valued processes, stochastic filtering and control, large deviations, and spatial point processes. Work in the area of stochastic analysis will focus on stochastic partial differential equations, the analysis of simulation schemes for stochastic equations, and the relationship between stochastic equations and martingale problems. Filtering equations for nonstandard observation structures will be derived and analyzed. Previous work on the martingale characterization of regular control problems will be extended to singular and partially observed control problems. Methods for the study of large deviations for Markov processes based on convergence of nonlinear semigroups will be extended to new classes of models. 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. In particular, filtering equations provide a method of processing information about phenomena that cannot be directly and/or accurately observed, by using a mathematical model for the relationship between the unobserved phenomena and quantities that can be observed. Examples of the kind of situations that will be considered include models in which the observed information is spatially distributed such as might be obtained from satellite observations and models of communications networks in which one attempts to obtain information about the state of the network from observations at a few points in the network.