Complex Stochastic Systems

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9971571Kurtz Research programs will be carried out in the areas of stochastic analysis, measure-valued processes, stochastic filtering and control, large deviations, and spatial point processes. Work in the area of stochastic analysis will focus on stochastic partial differential equations, the analysis of simulation schemes for stochastic equations, and the relationship between stochastic equations and martingale problems. Filtering equations for nonstandard observation structures will be derived and analyzed. Previous work on the martingale characterization of regular control problems will be extended to singular and partially observed control problems. Methods for the study of large deviations for Markov processes based on convergence of nonlinear semigroups will be extended to new classes of models. Spatial point processes characterized as stationary distributions of Markov birth and death processes and as solutions of stochastic equations driven by Poisson random measures will be studied. The work will focus on spatial ergodicity and central limit theorems with applications to parameter estimation in mind. 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, obtaining information about the true state of a system from corrupted or "noisy" observations, and determining how to influence or "control" the evolution of the models and the phenomena they represent. In particular, filtering equations provide a method of processing information about phenomena that cannot be directly and/or accurately observed, by using a mathematical model for the relationship between the unobserved phenomena and quantities that can be observed. Examples of the kind of situations that will be considered include models in which the observed information is spatially distributed such as might be obtained from satellite observations and models of communications networks in which one attempts to obtain information about the state of the network from observations at a few points in the network.