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 Title: Martingales in Filtering and Geometry Author(s): Bauer, Robert Otto Doctoral Committee Chair(s): Donald Burkholder Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics 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. Issue Date: 1997 Type: Text Language: English Description: 54 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997. URI: http://hdl.handle.net/2142/86944 Other Identifier(s): (MiAaPQ)AAI9737047 Date Available in IDEALS: 2015-09-28 Date Deposited: 1997
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