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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)

Abstract

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.

Keywords

Hilbert Space Markov Process Wiener Process Stochastic Partial Differential Equation Bilinear System 
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References

  1. [1]
    A. GERMANI, M. PICCIONI: Finite Dimensional Approximation for Stochastic Bilinear Differential Equations on Hilbert Spaces, Report R.61, IASI-CNR, 1983.Google Scholar
  2. [2]
    A. GERMANI, M. PICCIONI: A Galerkin Approximation for the Zakai Equation, Proceedings 11-th IFIP Conference on System Modelling and Optimization, Copenaghen, 1983, Springer-Verlag, to appear.Google Scholar
  3. [3]
    M. ZAKAI: On the Optimal Filtering of Diffusion Processes, Z. Wahrschein. verw. Geb., 11, 1969, pp. 230–243.Google Scholar
  4. [4]
    H. KUNITA: Cauchy Problem for Stochastic Partial Differential Equations arising in Nonlinear Filtering Theory, Systems & Control Letters, 1, 1981, pp. 37–41.Google Scholar
  5. [5]
    N.V. KRYLOV, B.L. ROZOVSKII: On the First Integral and Liouville Equations for Diffusion Processes, in Stochastic Differential Systems, Ed. M. Arato, D. Vermes, A.V. Balakrishman, Springer-Verlag, 1981.Google Scholar
  6. [6]
    E. PARDOUX: Stochastic Partial Differential Equations and Filtering of Diffusion Processes, Stochastics, 3, 1979, pp. 127–167.Google Scholar
  7. [7]
    M.H.A. DAVIS: On a Multiplicative Functional Transformation Arising in Nonlinear Filtering Theory, Z. Wahrsch. Verw. Geb., 54, 1980, pp. 125–139.Google Scholar
  8. [8]
    G. DA PRATO, M. IANNELLI, L. TUBARO: Linear Stochastic Differential Equations in Hilbert Spaces, Rend. Acc. Naz. Lincei, LXIV, 1978, pp. 22–29.Google Scholar
  9. [9]
    G. KOCH: Volterra Series Expansion for Stochastic Bilinear Systems, Report R. 2-07 of Istituto di Automatica, University of Rome, 1972.Google Scholar
  10. [10]
    G. KALLIANPUR: Stochastic Filtering Theory, Springer-Verlag, New York, 1980.Google Scholar
  11. [11]
    M. KAC: On Some Connections between Probability Theory and Differential and Integral Equations, Proc. 2nd Berkeley Symp. on Math. Stat. and Prob., 1951, pp. 189–215.Google Scholar
  12. [12]
    L. ARNOLD, W. KLIEMANN: Qualitative Theory of Stochastic Systems, in: Probabilistic Analysis and Related Topics, Vol. 3, Ed. by A.T. Bharucha Reid, Academic Press, New York, 1983.Google Scholar
  13. [13]
    T. KAILATH: Linear Systems, Prentice-Hall, Englewood Cliffs, 1980.Google Scholar
  14. [14]
    K. YOSIDA: Functional Analysis, Springer-Verlag, Berlin, 1980.Google Scholar

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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