Complex Stochastic Systems

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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.