Complex Stochastic Systems

技术描述。用于指定模型和理解模型的不同表述之间关系的工具对于模型开发和分析至关重要。这个项目的第一部分侧重于将模型指定为随机方程的解与将模型指定为鞅问题的解的更抽象的方法之间的关系。每种方法都有其优势，提供不同的直观洞察力和不同的分析方法。验证这两种方法的等价性可确保从每种方法中提取的方法应用于另一种方法的有效性。这两种方法共同提供了强大的方法，以确保模型是唯一确定的，并派生计算技术和模型简化。该项目的第二部分将开发新的方法来表示大粒子聚集的过程。这些粒子表示为分析和近似相关模型提供了强大的工具。该项目的一个目标是通过提供更直观的方法来构建表示和更清晰地识别构建中使用的构建块，使这些工具更容易被非专业人员使用。第三部分研究了具有时滞的随机控制问题。受控随机系统和对控制有延迟响应的系统基本上在每个应用领域都有出现。项目的这一部分关注的是将无延迟系统中易于理解的工具扩展到有延迟的系统。非技术描述。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。所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，以及构建针对特定科学应用的模型。该提案涉及生物学中的具体应用，而待开发的理论和方法在计算机科学和金融等其他领域也有应用。该项目还将为研究生和博士后研究人员提供肥沃的培训场地。特别是，对训练有素的数学科学家有很高的需求，他们必须有兴趣和专业知识，为解决生物学各个领域出现的问题做出贡献。

Technical description. Tools for specifying models and understanding the relationship between different formulations of a model are critical for model development and analysis. The first part of this project focuses on the relationship between specifying a model as a solution of a stochastic equation and the more abstract approach of specifying the model as a solution of a martingale problem. Each approach has its strengths providing different intuitive insight and different methods of analysis. Verifying the equivalence of these two approaches assures the validity of methods drawn from each applied to the other. Together, the two approaches provide powerful methodology for insuring that the models are uniquely determined and for deriving computational techniques and model simplifications. The second part of the project will develop new methods of representing processes in terms of large aggregations of ?particles.? These particle representations provide powerful tools for analyzing and approximating the associated models. One goal of the project will be to make these tools more readily available to the nonexpert by providing more intuitive approaches to the construction of the representations and clearer identification of the building blocks used in the construction. The third part of the proposal considers stochastic control problems with delay. Controlled random systems arise in essentially every area of application as do systems with delayed response to control. This part of the project is concerned with extending tools well understood in systems without delay to systems with delay.Non-technical description. 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 constructing models addressing specific scientific applications. The proposal addresses specific applications in biology and the theory and methodology to be developed have applications in other areas such as computer science and finance. The project will also provide a fertile training ground for graduate students and postdoctoral researchers. In particular, there is a high demand for well-trained mathematical scientists with the interest and expertise necessary to contribute to the solution of problems arising in all areas of biology.