Stochastic Control Problems and Numerical Methods

该方案涉及随机控制及其应用中的一系列关键问题，针对随机控制中的数值方法，目标是得到一大套有效的算法，这些算法编程方便，具有良好的数值特性，并且在实际条件下存在收敛定理，从而使数值方法能够发挥其作为研究和设计的实用而有效的方法的作用。对于遍历成本问题，将开发出具有更快收敛速度的算法。该方法将推广到随机微分对策中。这将允许处理健壮和风险敏感的问题，以及在许多问题中，噪声的强度受到控制的影响。我们将设计实用且性能良好的算法。数值算法一般假定漂移项是有界的，通信理论中存在无界漂移和可控漂移问题，相应的收敛算法将被开发出来。用单一控制来处理遍历成本是剩余的关键数值问题领域之一，并且在通信理论和重交通限制模型的应用中是至关重要的。将开发出性能良好的算法。更广泛地说，遍历标准和单一控制的繁忙交通限制的理论将完成。我们还将发展一套完整的近似方法理论，用于移动通信系统在信道以任意方式快速时变时的建模和控制。重交通模型和相关的极限定理为分析和控制这类系统提供了很好的指导。

The proposal deals with a large set of key problems in stochastic control and applications.Concerning Numerical Methods in Stochastic Control, the aim is to have a large set of effective algorithms which can be conveniently programmed, have good numerical properties, are robust, and for which there are convergence theorems under realistic conditions, so that numerical methods can fulfill their role as a practical and efficient approach to both investigation and design.The foundations are the Markov chain approximation methods. Algorithms with faster rate of convergence for the ergodic cost problem will be developed. The methods will be extended to stochastic differential games. This will allow treatment of robust and risk sensitive problems as well In many problems, the intensity of the noise is affected by the control. We will design practical and well performing algorithms. Numerical algorithms generally suppose that the drift terms are bounded.Problems with unbounded and controlled drift occur in communications theory and appropriate convergent algorithms will be developed. The treatment of ergodic costs with singular controls is one of the key remaining numerical problem areas, and is crucial in applications to communications theory and for heavy traffic limit models. Well performing algorithms will be developed. More generally, the theory of heavy traffic limits for the ergodic criteria and singular controls will be completed. We will also develop a complete theory of approximation methods for the modeling and control of Mobile communications systems when the channels are rapidly time varying in an arbitrary way. The heavy traffic models and associated limit theorems provide good guides for the analysis and control of a large variety of such systems.