Stochastic Control Problems and Numerical Methods

小行星9703895 随机系统中的一个广泛的主题将被处理。 1992年的《P.I.》(with P.Dupuis）是目前连续时间随机控制数值方法的主要来源，也是算法和收敛性证明的主要来源。 对于某些类型的非常感兴趣的问题，有公开可用的代码。 通信领域的实例充分说明了该理论和算法的巨大潜力。 一些重要的问题仍然存在。 其目的是发展一个高度可用的理论和一套技术，使数值方法可以取代它们作为一个非常实用和有效的方法，调查和设计。 该方法将被扩展到涵盖随机微分对策。 区域分解方法将开发的目的是促进解决问题的高维。 由于控制的存在，这些涉及的问题与文献中所做的完全不同。 高阶算法将被研究。 这里的主要困难是，经典的证明方法不适用，由于典型的退化和奇异性的问题。 需要使用概率方法，类似于用于处理主要数值算法的弱收敛方法。 高速通信中控制问题的研究将继续进行。 最近的工作已经证明了巨大的价值，交通量大的扩散近似方法，结合数值探索所考虑的问题。由于大量的独立用户，交通繁忙的控制问题的近似是可行的。 其目的是表明，简单的（限制/聚合）系统的好（或接近最优）的政策也是好的（或接近最优）的实际物理系统，从而简化设计和分析。 然后，开发收敛的数值方法，探索问题在许多不同的方式，获得信息的权衡在各种损失的功能的设计参数。 最近的工作沿着这些路线已经获得了重要的设计信息，这是不能得到的其他目前的方法。

9703895 Kushner A broad range of topics in stochastic systems will be treated. The 1992 book of the P.I. (with P.Dupuis) is now the primary source of numerical methods in continuous time stochastic control, as well as of algorithms and convergence proofs. There are publicly available codes for some classes of problems of great interest. The great potential of the theory and algorithms has been well illustrated by examples from communications. A number of important problems remain. The aim is the development of a highly usable theory and set of techniques, so that numerical methods can take their place as a very practical and efficient approach to both investigation and design. The methods will be extended to cover stochastic differential games. Domain decomposition methods will be developed with the aim of facilitating the solution of problems in high dimensions. Owing to the presence of the controls, these involve issues quite different from what has been done in the literature. Higher order algorithms will be investigated. The main difficulty here is that the classical methods of proof do not apply owing to the typical degeneracies and singularities in the problem. Probabilistic methods need to be used, analogous to the weak convergence methods used to deal with the main numerical algorithms. The work on control problems in high speed communications will be continued. The recent work has demonstrated the great value of the heavy traffic-diffusion approximation methods, when combined with numerical exploration for the problems considered. Due to the large number of independent users, heavy traffic approximations of the control problem is feasible. The aim is to show that good (or nearly optimal) policies for the simpler (limit/aggregated) system are also good (or nearly optimal) for the actual physical system, thereby simplifying design and analysis. One then develops convergent numerical methods to explore the problem in many different ways, obtaining information on the trad eoffs in the various losses as functions of the design parameters. Recent work along these lines has obtained information of importance to design which could not have been obtained by other current approaches.