Complex Stochastic Systems

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This research program will focus on singular martingale problems and stochastic differential equations, particle representations for stochastic partial differential equations and measure-valued processes, and spatial point processes. The work on singular martingale problems and related stochastic equations addresses fundamental questions regarding the specification, transformation and approximation of the models. Applications to controlled stochastic networks motivate much of the work. Particle representations provide a powerful tool for understanding the behavior of complex stochastic processes. Work to be performed will extend the range of applicability of these methods, develop new constructions of particle representations, and address problems of existence and uniqueness of solutions of the stochastic equations underlying these representations. Application of the results to filtering and to theoretical models in finance will provide both useful intuition and a check on the appropriateness of the theoretical developments. 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. Motivating examples include models for the effects on asset prices of the valuations assigned by a large number of traders and for communication and computer networks.