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https://hdl.handle.net/2142/86944
Description
Title
Martingales in Filtering and Geometry
Author(s)
Bauer, Robert Otto
Issue Date
1997
Doctoral Committee Chair(s)
Donald Burkholder
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
We give three applications of martingale theory. First, we study a problem in real-time target tracking. Realistic assumptions, namely limited processing power, turn the variance into a stochastic process. We transform and compensate the variance process so as to obtain a martingale. We find conditions on the parameters under our control that yield a satisfactory tracking mechanism. Specifying the relation between two quantities, we determine an optimal tracking procedure. Second, we give an explicit representation for the solution of the heat equation for trivial vector bundles using Ito's formula and an elementary martingale convergence result. Third, we give a martingale characterization of Yang-Mills fields. This uses stochastic analogues of lasso-forms and integrated lassos. In dimension 4, we relate the Yang-Mills action to the quadratic variation of the martingale used in the characterization. For the special case of self-dual Yang-Mills fields we give an energy identity.
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