Topics in Nonlinear Filtering Theory.
Abstract
This thesis studies two topics in the theory of nonlinear filtering, the use of multiple stochastic integrals to analyze filters, and the use of Lie algebraic and operator-theoretic techniques to discover new, finite-dimensionally solvable filtering problems. The main results of the multiple integral techniques are: (1) A simplier and more insightful proof of a result of S. Marcus on filtering polynomials functions of a Gauss-Markov process, (2) A formula for representing the product of two multiple integrals as a sum of multiple integrals, thus providing a rudimentary calculus of multiple integral expansions, (3) An expansion of the optimal means square filter as a ratio of two multiple integral expansions, and (4) Integral equations for the kernels of the best mean square filter of the class of (finite) r order multiple integral expansions. The problem of estimating a diffusion process observed in white noise is studied with Lie algebra techniques. Necessary conditions, and in the scalar case, necessary and sufficient conditions, are given for estimation algebra finite dimensionality. Examples of scalar problems with fin. dim. estimation algebras are discussed, and it is shown that, from among them, no new cases exist for which Zakai's equation can be solved by a Wei-Norman type method. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1980
- Accession Number
- ADA096973
Entities
People
- Daniel L. Ocone
Organizations
- Massachusetts Institute of Technology