Stochastic processes and applications

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I solve socially and economically important problems by developing novel models and algorithms. My past research has affected several companies positively, resulted in one patent award with three more filings (all by partner companies), earned me a consulting role with CBS show `Numbers', and won me the 2001 PIMS national outreach award. Future plans include studying climate change through ice core samples, improving the network by detecting threats and bottlenecks optimally, solving targeted advertizing by optimally selecting ads and classifying viewership and securing waters by tracking submarines and pollution. Some mathematical novelties associated with these goals include developing filtering theories that can handle heavy-tailed processes, support robust implementations beyond those already established, estimate signals from mere (partial-differential-equation-defined) traces and observations that are defined in terms of complicated point processes, Markov chains and random environments instead of the usually distorted signal in presence of noise setting. In every application, I work with corporations or individuals dedicated to the area of interest.   For fundamentally mathematical problems, I will utilize homeomorphic methods Prof. Blount and I just developed with other methods to settle basic probabilistic problems such as existence to martingale problems, tightness, exponential tightness, weak convergence, large deviations and long time behaviour for a large class of stochastic models. Indeed, we have already completed a study on tightness using these homeomorphic methods and the results were amazing: the proofs were shorter while the results were far more general, allowing greater applicability, than the standard results in Ethier and Kurtz (1986) or Jakubowski (1986). In another vein, I will further develop the `look down' particle representation, introduced by Professors Kurtz, Donnelly and Xiong, and investigate its application to stochastic partial differential equations, measure-valued branching processes, population models, controlled Markov process and filtering problems. Notwithstanding Kurtz's 1998 paper, the general particle representation theorem has not been established.