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Geometric Theory of Nonlinear Filtering
MASSACHUSETTS INST OF TECH CAMBRIDGE DEPT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE
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Until quite recently, the basic approach to non-linear filtering theory was via the innovations method, originally proposed by Kailath ca. 1967 and subsequently rigorously developed by Fujisaki, Kallianpur and Kunita 1 in their seminal paper of 1972. The difficulty with this approach is that the innovations process is not, in general, explicitly computable excepting in the well-known Kalman-Bucy case. To circumvent this difficulty, it was independently proposed by Brockett-Clark 2, Brockett 3, Mitter 4 that the construction of the filter be divided into two parts i a universal filter which is the evolution equation describing the unnormalized conditional density, the Duncan-Mortensen- Zakai D-M-Z equation and ii a state-output map, which depends on the statistic to be computed, where the state of the filter is the unnormalized conditional density. The reason for focusing on the D-M-Z equation is that it is an infinite-dimensional bi-linear system driven by the incremental observation process, and a much simpler object than the conditional density equation which is a non-linear equation and can be treated using geometric ideas. Moreover, it was noticed by this author that this equation bears striking similarities to the equations arising in Euclidean-quantum mechanics and it was felt that many of the ideas and methods used there could be used in this context. The ideas and methods referred to here are the functional integration view of Feynman for a modern exposition see Glimm-Jaffe 5.
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