Complex Stochastic Systems

9626116个库尔茨抽象研究计划将在马尔可夫过程、空间点过程、随机分析和随机控制领域进行。关于马尔可夫过程的工作将集中在与种群遗传学和其他种群模型以及马尔可夫过程的新型随机方程有关的测值过程上。将开发空间点过程的新模型，并处理相应的统计问题。以前关于由半鞅随机测度驱动的随机方程的工作将被扩展，特别强调在随机网络的研究中产生的无限方程组和模型。一般随机控制问题的理论发展将包括最优奇异控制的逼近方法和存在定理。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，并确定如何影响或“控制”模型及其所代表的现象的演变。我们将研究几种特殊类型的模型。

9626116 Kurtz ABSTRACT Research programs will be carried out in the areas of Markov processes, spatial point processes, stochastic analysis, and stochastic control. Work on Markov processes will focus on measure-valued processes related to population genetics and other population models and new types of stochastic equations for Markov processes. New models for spatial point processes will be developed and corresponding statistical issues addressed. Previous work on stochastic equations driven by semimartingale random measures will be extended with particular emphasis on infinite systems of equations and models arising in the study of stochastic networks. Theoretical developments for general stochastic control problems will include approximation methodology and existence theorems for optimal singular controls. 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, and determining how to influence or "control" the evolution of the models and the phenomena they represent. Several particular types of models will be studied.