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Convergence of implicit discretization schemes for linear differential equations with application to filtering

Part of the Lecture Notes in Mathematics book series (LNM,volume 1236)

Keywords

  • Weak Convergence
  • Infinitesimal Generator
  • Contraction Semigroup
  • Martingale Problem
  • Integrable Martingale

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References

  1. J. S. Baras, A. La Vigna, "Expert systems and VLSI architectures for real-time non-Gaussian detectors and filters", in C. Byrnes and A. Lindquist eds., Proceedings of MTNS-85, North-Holland, to appear.

    Google Scholar 

  2. Ya. I. Belopol'skaya, Z. I. Nagolkina, "On a class of stochastic partial differential equations," Th. of Prob. and its Appl., 27, 592–599, 1982.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. P. Billingsley, Convergence of Probability Measures, John Wiley, New York, 1968.

    MATH  Google Scholar 

  4. P. Billingsley, Weak Convergence of Measures: Applications in Probability, SIAM, Philadelphia, 1971.

    CrossRef  MATH  Google Scholar 

  5. J. M. C. Clark, "The design of robust approximations to the stochastic differential equations of nonlinear filtering," in J. K. Skwirzynski ed., Communication Systems and Random Process Theory, Sijthoff and Noordhoff, Aalpen aan den Rijn, 1978.

    Google Scholar 

  6. E. B. Dynkin, Markov Processes — I, Springer-Verlag, Berlin, 1965.

    CrossRef  MATH  Google Scholar 

  7. A. Germani, M. Piccioni, "A Galerkin approximation for the Zakai equation," in P. Thoft-Christensen, ed., Systems Modelling and Optimization, Springer-Verlag, Berlin, 1984.

    Google Scholar 

  8. I. I. Gihman, A. V. Skorohod, The Theory of Stochastic Processes, III, Springer-Verlag, New York, 1979.

    CrossRef  MATH  Google Scholar 

  9. E. Hille, Functional Analysis and Semigroups, AMS, New York, 1948.

    MATH  Google Scholar 

  10. J. Jacod, Calcul Stochastique et Problémes de Martingales, Springer-Verlag, Berlin, 1977.

    MATH  Google Scholar 

  11. G. Kallianpur, C. Striebel, "Estimation of stochastic systems: arbitrary system process with additive white noise observation errors," Ann. Math. Stat., 39, 785–801, 1968.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976.

    CrossRef  MATH  Google Scholar 

  13. T. G. Kurtz, "Extensions of Trotter's operator semigroup approximation theorems," J. Funct. Anal., 3, 354–375, 1969.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. T. G. Kurtz, "Semigroups of conditional shifts and approximations of Markov processes." Ann. Prob., 4, 618–642, 1975.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. T. G. Kurtz, Approximation of Population Processes, SIAM, Philadelphia, 1981.

    CrossRef  MATH  Google Scholar 

  16. H. J. Kushner, Probability Methods for Approximation in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977.

    MATH  Google Scholar 

  17. H. J. Kushner, "A robust discrete state approximation to the optimal nonlinear filter for a diffusion," Stochastics, 3, 75–83, 1979.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, MIT, Cambridge, 1984.

    MATH  Google Scholar 

  19. H. J. Kushner, H. Huang, "Approximate and limit results for nonlinear filters with wide bandwidth observation noise," Report LCDS 84-36, 1984.

    Google Scholar 

  20. F. Legland, "Estimation de paramètres dans les processus stochastiques en observation incomplète," Thèse, Université Paris IX, 1981.

    Google Scholar 

  21. T. Lindvall, "Weak convergence of probability measures and random functions in the function space D[0,)," J. Appl. Prob., 10, 109–121, 1973.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. P.-A. Meyer, Martingales and Stochastic Integrals I, Springer-Verlag, Berlin, 1978.

    Google Scholar 

  23. M. Metivier, Semimartingales, W. de Gruyter, Berlin, 1982.

    CrossRef  MATH  Google Scholar 

  24. R. D. Richtmyer, K. W. Morton, Difference Methods for Initial-Value Problems, Interscience, New York, 1967.

    MATH  Google Scholar 

  25. B. Stewart, "Generation of analytic semigroups by strongly elliptic operators," Trans. Amer. Math. Soc., 199, 141–162, 1974.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. D. W. Stroock, S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979.

    MATH  Google Scholar 

  27. H. Trotter, "Approximation of semigroups of operators," Pacific J. Math., 8, 887–919, 1958.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. H. Trotter, "On the product of semigroups of operators," Proc. Amer. Math. Soc., 10, 545–551, 1959.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. D. Williams, Diffusions, Markov Processes and Martingales, Vol. 1, John Wiley, Chichester, 1979.

    MATH  Google Scholar 

  30. K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1968.

    CrossRef  MATH  Google Scholar 

  31. M. Zakai, "On the optimal filtering of diffusion processes," Z. Wahr. verw. Geb., 11, 230–243, 1969.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1987 Springer-Verlag

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Piccioni, M. (1987). Convergence of implicit discretization schemes for linear differential equations with application to filtering. In: Da Prato, G., Tubaro, L. (eds) Stochastic Partial Differential Equations and Applications. Lecture Notes in Mathematics, vol 1236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072892

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  • DOI: https://doi.org/10.1007/BFb0072892

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