Nonlinear filtering for markov processes: An L2 approach

  • A. Germani
  • M. Piccioni
Session 10 Filtering
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 62)


In this paper the Zakai equation of nonlinear filtering is directly derived as a mild stochastic differential equation on a Hilbert space. This is established when the state process is Markov, with a generator on some L2 space, and the observation process is corrupted by white noise. The main step is the derivation of a Feynman-Kac like formula for mild stochastic differential equations. In such a way well-known results in literature are generalized and at the same time their proofs are made much more simpler; moreover the Hilbert space setting is the most appropriate for approximation procedures. A final example is given in which the Zakai equation is written for a stochastic differential system in which the state equation is linear with an arbitrary number of noises, partly overcoming one of the up to now most serious limitations of such theory.


Hilbert Space Markov Process Wiener Process Stochastic Partial Differential Equation Bilinear System 
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Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • A. Germani
    • 1
  • M. Piccioni
    • 2
  1. 1.Istituto di Analisi dei Sistemi ed Informatica del C.N.R.RomaItaly
  2. 2.Istituto di MatematicaInformatica e Sistemistica dell'Università di UdineUdineItaly

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