On the joint nonlinear filtering-smoothing of diffusion processes

The smoothing of diffusions d x t = f( x t ) d t + σ( x t ) d w t , measured by a noisy sensor d y t = h( x t ) d t + d v t , where w t and v t are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of ( x τ x t ), 0 ≤ τ < t, conditioned on t...

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Bibliographic Details
Published in:Systems & control letters 1986-07, Vol.7 (4), p.317-321
Main Authors: Zeitouni, O., Bobrovsky, B.Z.
Format: Article
Language:English
Subjects:
Applied sciences
Computer science
control theory
systems
Control theory. Systems
Exact sciences and technology
Finite-dimensional filters
Modelling and identification
Nonlinear filtering
Smoothing
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Summary:The smoothing of diffusions d x t = f( x t ) d t + σ( x t ) d w t , measured by a noisy sensor d y t = h( x t ) d t + d v t , where w t and v t are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of ( x τ x t ), 0 ≤ τ < t, conditioned on the observation path { y s , 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of x τ alone, a set of ‘forward and reverse’ equations needs to be solved.
ISSN:0167-6911
1872-7956
DOI:10.1016/0167-6911(86)90046-0