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Sequential Identification of Linear Dynamic Systems with Memory

  • Uwe Küchler
  • Vyacheslav Vasiliev
Article

Abstract

This paper presents a sequential estimator for some unknown parameters in stochastic linear systems with memory. As examples stochastic differential equations with time delayed drift are considered. Based on the maximum likelihood method, we construct an estimation procedure with given accuracy in the sense of the L p -norm (p≥ 2). It is shown, that this procedure works also in certain cases, when the normalized information matrix of the observed process is asymptotically degenerated. The almost surely consistency of the proposed estimators and the asymptotic behavior of the length of observations are derived.

stochastic differential equations with memory time delay maximum likelihood estimator sequential analysis guaranteed accuracy 

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References

  1. Galtchouk, L. and Konev, V.: On sequential estimation of parameters in semimartingale regression models with continuous time parameter, Ann. Statist. 29(5) (2001), 1508–1536.Google Scholar
  2. Gushchin, A.A.and Küchler, U.: Asymptotic inference for a linear stochastic differential equation with time delay, Bernoulli 5(6) (1999), 1059–1098.Google Scholar
  3. Gushchin, A.A.and Küchler, U.: On stationary solutions of delay differential equations driven by a Lèvy process, Stoch. Proc. Appl. 88 (2000), 195–211.Google Scholar
  4. Hale, J. K. and Verduyn Lunel, S. M.: Introduction to Functional-Differential Equations, Springer-Verlag, New York, 1993.Google Scholar
  5. Kolmanovskii, V. and Myshkis, A.: Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992.Google Scholar
  6. Konev, V. V. and Pergamenshchikov, S. M.: Sequential estimation of the parameters of diffusion processes, Probl. Inform. Trans. 21(1) (1985), 48–62.Google Scholar
  7. Konev, V. V. and Pergamenshchikov, S. M.: Sequential estimation of the parameters of unstable dynamical systems in continuous time, In: F. Tarasenko (ed), Mathematical Statistics and Applications, Publishing House of Tomsk University, Tomsk, 11 (1987), 85–94.Google Scholar
  8. Konev, V. V. and Pergamenshchikov, S. M.: Sequential estimation of the parameters of linear unstable stochastic systems with guaranteed accuracy, Probl. Inform. Trans. 28(4) (1992), 35–48.Google Scholar
  9. Konev, V. and Vasiliev, V.: Scientific computation, modelling and applied mathematics, In: Proceedings of the 15th IMACS World Congress, 24–29 August 1997, Berlin, Germany, 5 (1997), 87–91.Google Scholar
  10. Küchler, U. and Kutoyants, Yu. A.: Delay estimation for stationary diffusion-type processes, Scand. J. Statist. 27(3) (2000), 405–414.Google Scholar
  11. Küchler, U. and Mensch, B.: Langevins stochastic differential equations extended by a time-delayed term, Stoch. Stoch. Rep. 40 (1991), 23–42.Google Scholar
  12. Küchler, U. and Soerensen, M.: Exponential Families of Stochastic Processes, Springer-Verlag, New York, Heidelberg, 1997.Google Scholar
  13. Küchler, U. and Vasiliev, V.: On sequential parameter estimation for some linear stochastic differential equations with time delay, Sequent. Anal. 20(3) (2001), 117–146.Google Scholar
  14. Liptser, R. S. and Shiryaev, A. N.: Statistics of Random Processes, Springer-Verlag, New York, Heidelberg, 1977.Google Scholar
  15. Liptser, R. S. and Shiryaev, A. N.: Theory of Martingales, Kluwer, Dordrecht, Boston, London, 1989.Google Scholar
  16. Mao, X.: Stochastic Differential Equations and Application, Harwood Publishing, Chichester, 1997.Google Scholar
  17. Mohammed, S. E.-A.: Stochastic differential systems with memory: theory, examples and applications, In: Progress in Probability and Statistics, Birkhäuser, 42 (1984), 1–77.Google Scholar
  18. Mohammed, S. E.-A. and Scheutzow, M. K. R.: Lyapunov exponents and stationary solutions for affine stochastic delay equations, Stoch. Stoch. Rep. 29 (1990), 259–283.Google Scholar
  19. Novikov, A. A.: The sequential parameter estimation in the process of diffusion type, Probab. Theory Appl. 16(2) (1971), 394–396.Google Scholar
  20. Putschke, M. U.: Affine stochastische Funktional differentialgleichungen und lokal asymptotische Eigenschaften ihrer Parameterschätzungen, Humboldt Universität zu Berlin, Berlin, 2001.Google Scholar
  21. Reiß, M.: Nonparametric Estimation for Stochastic Delay Differential Equations, Dissertation, Institut für Mathematik, Humboldt Universität zu Berlin, 2002.Google Scholar
  22. Vasiliev, V. A. and Konev, V. V.: On sequential identification of linear dynamic systems in continuous time by noisy observations, Probl. Contr. Inform. Theory 16(2) (1987), 101–112.Google Scholar
  23. Vasiliev, V. A. and Konev, V. V.: On sequential parameter estimation of continuous dynamic systems by discrete time observations, Probl. Contr. Inform. Theory 19(3) (1990), 197–207.Google Scholar

Copyright information

© Kluwer Academic Publishers 2005

Authors and Affiliations

  • Uwe Küchler
    • 1
  • Vyacheslav Vasiliev
    • 2
  1. 1.Institute of MathematicsHumboldt University BerlinBerlinGermany
  2. 2.Department of Applied Mathematics and CyberneticsTomsk State UniversityTomskRussia

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