Stochastic Control Problems and Numerical Methods

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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.