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Stochastic filtering theory: A discussion of concepts, methods, and results

  • J. H. van Schuppen
Part I: Survey Lectures
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 16)

Keywords

Markov Process Stochastic Differential Equation Stochastic Control Jump Process Stochastic Dynamical System 
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References

  1. [1]
    R. BOEL, P. VARAIYA, E. WONG, Martingales on Jump Processes. I. Representation Results. II. Applications, SIAM J. Control, 13, (1975), pp. 999–1061.Google Scholar
  2. [2]
    P.M. BRÉMAUD, A martingale approach to point processes, Electronics Research Lab., Memo M-345, University of California, Berkeley, Cal., 1972.Google Scholar
  3. [3]
    P.M. BRÉMAUD, The martingale theory of point processes over the real half line admitting an intensity, Control Theory, Numerical Methods and Computer Systems Modelling, Lect. Notes in Econ. and Math. Systems Th., vol. 107, Springer-Verlag, Berlin, 1974, pp. 519–542.Google Scholar
  4. [4]
    P.M. BRÉMAUD, La méthode des semi-martingales en filtrage lorsque l'observation est un processus ponctuel marqué, Séminaire de Probilités, Lect. Notes in Math., 511, Springer-Verlag, Berlin, 1975, pp 1–18.Google Scholar
  5. [5]
    P.M. BRÉMAUD, M. YOR, Changes of filtration and of probability measures, report IRIA.Google Scholar
  6. [6]
    P. BREMAUD, J. JACOD, Processus Ponctuels et Martingales: Résultats récents sur la modélisation et le filtrage, Advances in Appl. Probability, 9, (1977), pp. 362–416.Google Scholar
  7. [7]
    C. DELLACHERIE, P.A. MEYER, Probabilités et Potentiel, Hermann, Paris, 1975.Google Scholar
  8. [8]
    P. FISHMAN, D. SNYDER, How to track a swarm of fireflies by observing their flashes, IEEE Trans. Information Theory, 21, (1975), pp. 692–694.Google Scholar
  9. [9]
    M. FUJISAKI, G. KALLIANPUR, H. KUNITA, Stochastic Differential Equations for the non-linear filtering problem, Osaka J. Math., 9, (1972). pp. 19–40.Google Scholar
  10. [10]
    J. JACOD, Un théorème de représentation pour les martingales discontinues, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 34, (1976), pp. 225–244.Google Scholar
  11. [11]
    T. KAILATH, The innovations approach to detection and estimation theory, Proc. IEEE, 58, (1970), pp. 680–695.Google Scholar
  12. [12]
    T. KAILATH, A view of three decades of linear filtering theory, IEEE Trans. Information Theory, 20, (1974), pp. 146–181.Google Scholar
  13. [13]
    G. KALLIANPUR, Non linear filtering, in Optimizing methods in Statistics, J.S. Rustagi ed., Acad. Press, N.Y., 1971, pp. 211–232.Google Scholar
  14. [14]
    R.E. KALMAN, A new approach to linear filtering and prediction problems, Trans. ASME, series D., Journal of Basic Engineering, 82, (1960), pp. 35–45.Google Scholar
  15. [15]
    R.E. KALMAN, R.S. BUCY, New results in linear filtering and prediction theory, Trans. ASME, series D., Journal of Basic Engineering, 83, (1961), pp. 95–108.Google Scholar
  16. [16]
    R.E. KALMAN, Linear stochastic filtering theory — reappraisal and outlook, Proc. Symp. System Theory, J. Fox ed., Polytechnic Press, N.Y., 1965, pp. 197–205.Google Scholar
  17. [17]
    R.E. KALMAN, P.L. FALB, M.A. ARBIB, Topics in Mathematical System Theory, McGraw-Hill, N.Y., 1969.Google Scholar
  18. [18]
    H. KUNITA, Asymptotic behaviour of the non-linear filtering errors of Markov processes, J. Multivariate Anal., 1, (1971), pp. 365–393.Google Scholar
  19. [19]
    H.J. KUSHNER, Dynamical equations for optimal nonlinear filtering, J. Differential Equations, 3, (1967), pp. 179–190.Google Scholar
  20. [20]
    A. LINDQUIST, G. PICCI, On the stochastic realization problem, SIAM J. Control, to appear.Google Scholar
  21. [21]
    A. LINDQUIST, G. PICCI, A state-space theory for stationary stochastic processes, Proc. 21st Midwest Symposium on Circuits and Systems, August 1978.Google Scholar
  22. [22]
    A. LINDQUIST, G. PICCI, G. RUCKEBUSCH, On minimal splitting subspaces and Markovian representations, preprint.Google Scholar
  23. [23]
    R.S. LIPTSER, A.N. SHIRYAYEV, Statistics of random processes. I. General Theory, II. Applications, Springer-Verlag, Berlin, 1977, 1978.Google Scholar
  24. [24]
    J.T. LO, A.S. WILLSKY, Estimation for rotational processes with one degree of freedom, IEEE Tans. Automatic Control, 20, (1975), pp. 10–33.Google Scholar
  25. [25]
    S.I. MARCUS, A.S. WILLSKY, Algebraic structure and finite dimensional nonlinear estimation, SIAM J. Math. Anal., 9, (1978), pp. 312–327.Google Scholar
  26. [26]
    H.P. McKEAN, Brownian motion with a several dimensional time, Theory Prob. Appl., 8, (1963), pp. 335–354.Google Scholar
  27. [27]
    P.A. MEYER, Probabilités et Potentiel, Hermann, Paris, 1965; english translation, Probability and Potential, Blaisdell, Waltham, Mass., 1966.Google Scholar
  28. [28]
    P.A. MEYER, un cours sur les intégrales stochastiques, in Séminaire de Probabilités X, Lecture Notes in Math., 511, Springer-Verlag, Berlin, 1975.Google Scholar
  29. [29]
    G. PICCI, Stochastic Realization of Gaussian processes, Proc. IEEE, 64, (1976), pp. 112–122.Google Scholar
  30. [30]
    M. RUDEMO, Doubly stochastic Poisson processes and process control, Acvances in Appl. Probability, 4, (1972), pp. 318–338.Google Scholar
  31. [31]
    M. RUDEMO, State estimation for partially observed Markov chains, J. Math. Anal. Appl., 44, (1973), pp. 581–611.Google Scholar
  32. [32]
    A. SEGALL, A martingale approach to modelling, estimation, and detection of jump processes, Ph. D. thesis, Stanford University, 1973.Google Scholar
  33. [33]
    A. SEGALL, M.H.A. DAVIS, T. KAILATH, Nonlinear filtering with counting observations, IEEE Trans. Information Theory, 21, (1975), pp. 143–149.Google Scholar
  34. [34]
    A. SEGALL, dynamic file assignment in a computer network, IEEE Trans. Automatic Control, 21, (1976), pp. 161–173.Google Scholar
  35. [35]
    A.N. SHIRYAYEV, Stochastic equations of nonlinear filtering of Markovian jump processes, Problems of Information Transmission, 2, (1966), pp. 1–18.Google Scholar
  36. [36]
    M. SKIBINSKY, Adequate subfields and sufficiency, Ann. Math. Stat., 38, (1967), pp. 155–161.Google Scholar
  37. [37]
    D.L. SNYDER, Filtering and detection for doubly stochastic Poisson processes, IEEE Trans. Information Theory, 18, (1972), pp. 97–102.Google Scholar
  38. [38]
    D.L. SNYDER, Random Point processes, Wiley, N.Y., 1975.Google Scholar
  39. [39]
    R.L. STRATONOVICH, Conditional Markov processes, Theor. Probability Appl., 5, (1960), pp. 156–178.Google Scholar
  40. [40]
    J. SZPIRGLAS, G. MAZZIOTTO, Modèle général de filtrage non linéaire et équations différentielles stochastiques associées, C.R. Acad. Sc. Paris, Série A, 286, (1978), pp. 1067–1070.Google Scholar
  41. [41]
    M.V. VACA, D.L. SNYDER, Estimation and Decision for observations derived from martingales: Part 1, Representations, IEEE Trans. Information Theory, 22, (1976), pp. 691–707.Google Scholar
  42. [42]
    J.H. VAN SCHUPPEN, Estimation Theory for continuous-time processes, a martingale approach, Ph.D. thesis, University of California, Berkeley, Memo ERL M-405, 1973.Google Scholar
  43. [43]
    J.H. VAN SCHUPPEN, Filtering, prediction, and smoothing for counting process observations, a martingale approach, SIAM J. Appl. Math., 32, (1977), pp. 552–570.Google Scholar
  44. [44]
    N. WIENER, The extrapolation, interpolation, and smoothing of stationary time series with engineering applications, Wiley, N.Y., 1949.Google Scholar
  45. [45]
    E. WONG, Stochastic Processes in Information and Dynamical Systems, McCraw-Hill, N.Y., 1971.Google Scholar
  46. [46]
    W.M. WONHAM, Some applications of stochastic differential equations to optimal nonlinear filtering, SIAM J. Control, 2, (1965), pp. 347–369.Google Scholar
  47. [47]
    A.I. YASHIN, Filtering of jump processes, Automat. Remote Control, 31, (1970), pp. pp. 725–730.Google Scholar
  48. [48]
    M. YOR, Sur les théories du filtrage et de la prédiction, in Séminaire de Probabilités XI, Lecture Notes in Math., 581, Springer-Verlag, Berlin, 1977, pp. 257–297.Google Scholar
  49. [49]
    M. ZAKAI, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 11, (1969), pp.230–243.Google Scholar
  50. [50]
    C. DOLÉANS-DADE, P.A. MEYER, Equations différentielles stochastiques, in Séminaire de Probabilités XI, Lecture Notes in Math., 581, Springer-Verlag, Berlin, 1977, pp. 376–382.Google Scholar
  51. [51]
    P.A. FROST, Examples of linear solutions to nonlinear estimation problems, Proc. 5th Princeton Conf. on Info. Sc. and Systems, 1971, pp. 20–24.Google Scholar
  52. [52]
    H. RAIFFA, R. SCHLAIFER, Applied Statistical Decision Division of Research, Harvard University, Boston, Mass., 1961.Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • J. H. van Schuppen
    • 1
  1. 1.Stichting Mathematisch CentrumAmsterdamThe Netherlands

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