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General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes

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Statistics of Random Processes

Part of the book series: Applications of Mathematics ((SMAP,volume 5))

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Abstract

Let (Ω, F, P) be a complete probability space, and let (F t ), 0 ≤ t ≤ T, be a nondecreasing family of right continuous σ-algebras of F augmented by sets from F of zero probability.

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Bibliography

Notes and References. 1

  1. Kolmogorov, A.N. (1941): Interpolation and extrapolation of stationary random sequences. Izv. Akad. Nauk SSSR, Ser. Mat., 5, 5

    MathSciNet  Google Scholar 

  2. Wiener, N. (1949): Extrapolation, Interpolation and Smoothing of Stationary Time Series. Wiley, New York

    Google Scholar 

  3. Yaglom, A.M. (1952): Introduction to the theory of stationary random functions. Usp. Mat. Nauk, 7, 5, 3–168

    MathSciNet  MATH  Google Scholar 

  4. Rozanov, Yu.A. (1967): Stationary Random Processes. Holden-Day, San Francisco

    MATH  Google Scholar 

  5. Rozovskii, B.L. (1972): Stochastic partial differential equations arising in nonlinear filtering problems. Usp. Mat. Nauk SSSR, 27, 3, 213–4

    MathSciNet  MATH  Google Scholar 

  6. Rozovskii, B.L. (1990): Stochastic Evolution Systems. Linear Theory and Application to Nonlinear Filtering. Reidel, Dordrecht

    Google Scholar 

  7. Stratonovich, R.L. (1960): Conditional Markov processes. Teor. Veroyatn. Primen., 5, 2, 172–95

    MATH  Google Scholar 

  8. Stratonovich, R.L. (1966): Conditional Markov Processes and their Applications to Optimal Control Theory. Izd. MGU, Moscow

    Google Scholar 

  9. Wentzell, A.D. (1965): On equations of the conditional Markov processes. Teor. Veroyatn. Primen., 10, 2, 390–3

    Google Scholar 

  10. Wonham, W.M. (1965): Some applications of stochastic differential equations to optimal nonlinear filtering. SIAM J. Control Optimization, 2, 347–69

    MathSciNet  MATH  Google Scholar 

  11. Kushner, H.J. (1964): On the dynamical equations of conditional probability density functions with applications to optimal stochastic control theory. J. Math. Anal. Appl., 8, 332–44

    Google Scholar 

  12. Kushner, H.J. (1967): Dynamical equations for nonlinear filtering. J. Differ. Equations, 3, 179–90

    Article  MathSciNet  MATH  Google Scholar 

  13. Shiryaev, A.N. (1966): On stochastic equations in the theory of conditional Markov processes. Teor. Veroyatn. Primen., 11, 1, 200–6

    Google Scholar 

  14. Shiryaev, A.N. (1966): Stochastic equations of nonlinear filtering of jump Markov processes. Probi. Peredachi Inf., 2, 3, 3–22

    Google Scholar 

  15. Shiryaev, A.N. (1970): Sur les Equations Stochastiques aux Dérivées Partielles. Actes Congrès Int. Math.

    Google Scholar 

  16. Liptser, R.S. and Shiryaev, A.N. (1968): Nonlinear filtering of diffusion type Markov processes. Tr. Mat. Inst. Steklova, 104, 135–80

    Google Scholar 

  17. Liptser, R.S. and Shiryaev, A.N. (1968): On the case of effective solution of the problems of optimal nonlinear filtering, interpolation, and extrapolation. Teor. Veroyatn. Primen., 13, 3, 570–1

    Google Scholar 

  18. Liptser, R.S. and Shiryaev, A.N. (1968): Nonlinear interpolation of the components of diffusion type Markov processes (forward equations, effective formulae). Teor. Veroyatn. Primen., 13, 4, 602–20

    MathSciNet  Google Scholar 

  19. Liptser, R.S. and Shiryaev, A.N. (1969): Interpolation and filtering of the jump component of a Markov process. Izv. Akad. Nauk SSSR, Ser. Mat., 33, 4, 901–14

    MathSciNet  MATH  Google Scholar 

  20. Liptser, R.S. (1967): On filtering and extrapolation of the components of diffusion type Markov processes. Teor. Veroyatn. Primen., 12, 4, 754–6

    MathSciNet  Google Scholar 

  21. Liptser, R.S. (1968): On extrapolation and filtering of some Markov processes, II. Kibernetika (Kiev), 6, 70–6

    Google Scholar 

  22. Kailath, T. (1968): An innovation approach to least-squares estimation, Parts I, II. IEEE Trans. Autom. Control, AC-13, 646–60.

    Google Scholar 

  23. Kailath, T. and Greesy, R. (1971): An innovation approach to least-squares estimation, Part IV. IEEE Trans. Autom. Control, AC-16, 720–27

    Google Scholar 

  24. Frost P. and Kailath, T. (1971): An innovation approach to least-squares estimation, Part III. IEEE Trans. Autom. Control, AC-16, 217–26

    Google Scholar 

  25. Striebel, C.T. (1965): Partial differential equations for the conditional distribution of a Markov process given noisy observations. J. Math. Anal. Appl., 11, 151–9

    Google Scholar 

  26. Yershov, M.P. (1969): Nonlinear filtering of Markov processes. Teor. Veroyatn. Primen., 14, 4, 757–8

    Google Scholar 

  27. Yershov, M.P. (1970): Sequential estimation of diffusion processes. Teor. Veroyatn. Primen., 15, 4, 705–17

    Google Scholar 

  28. Kallianpur, G. and Striebel, C. (1968): Estimation of stochastic systems: arbitrary system process with additive noise observation errors. Ann. Math. Stat., 39, 785–801

    Article  MathSciNet  MATH  Google Scholar 

  29. Kallianpur, G. and Striebel, C. (1969): Stochastic differential equations occurring in the estimation of continuous parameter stochastic processes. Teor. Veroyatn. Primen., 14, 4, 597–622

    MathSciNet  MATH  Google Scholar 

  30. Grigelionis, B. (1972): On stochastic equations of nonlinear filtering of random processes. Litov. Mat. Sb., 12, 4, 37–51

    MathSciNet  MATH  Google Scholar 

  31. Fujisaki, M., Kallianpur, G. and Kunita, H. (1972): Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math., 9, 1, 19–40

    MathSciNet  MATH  Google Scholar 

  32. Stratonovich, R.L. (1960): Conditional Markov processes. Teor. Veroyatn. Primen., 5, 2, 172–95

    MATH  Google Scholar 

  33. Stratonovich, R.L. (1966): Conditional Markov Processes and their Applications to Optimal Control Theory. Izd. MGU, Moscow

    Google Scholar 

  34. Shiryaev, A.N. (1966): On stochastic equations in the theory of conditional Markov processes. Teor. Veroyatn. Primen., 11, 1, 200–6

    Google Scholar 

  35. Liptser, R.S. and Shiryaev, A.N. (1968): Nonlinear filtering of diffusion type Markov processes. Tr. Mat. Inst. Steklova, 104, 135–80

    Google Scholar 

  36. Stratonovich, R.L. (1966): Conditional Markov Processes and their Applications to Optimal Control Theory. Izd. MGU, Moscow.

    Google Scholar 

  37. Liptser, R.S. and Shiryaev, A.N. (1968): On filtering, interpolation, and extrapolation of diffusion type Markov processes with incomplete data. Teor. Veroyatn. Primen., 13, 3, 569–70

    Google Scholar 

  38. Liptser, R.S. and Shiryaev, A.N. (1968): Extrapolation of multivariate Markov processes with incomplete data. Teor. Veroyatn. Primen., 13, 1, 17–38

    Google Scholar 

  39. Liptser, R.S. and Shiryaev, A.N. (1968): On the case of effective solution of the problems of optimal nonlinear filtering, interpolation, and extrapolation. Teor. Veroyatn. Primen., 13, 3, 570–1

    Google Scholar 

  40. Liptser, R.S. and Shiryaev, A.N. (1968): Nonlinear interpolation of the components of diffusion type Markov processes (forward equations, effective formulae). Teor. Veroyatn. Primen., 13, 4, 602–20

    MathSciNet  Google Scholar 

  41. Liptser, R.S. and Shiryaev, A.N. (1969): Interpolation and filtering of the jump component of a Markov process. Izv. Akad. Nauk SSSR, Ser. Mat., 33, 4, 901–14

    MathSciNet  MATH  Google Scholar 

  42. Liptser, R.S. (1967): On filtering and extrapolation of the components of diffusion type Markov processes. Teor. Veroyatn. Primen., 12, 4, 754–6

    MathSciNet  Google Scholar 

  43. Liptser, R.S. (1968): On extrapolation and filtering of some Markov processes, I. Kibernetika (Kiev), 3, 63–70

    Google Scholar 

  44. Liptser, R.S. (1968): On extrapolation and filtering of some Markov processes, II. Kibernetika (Kiev), 6, 70–6

    Google Scholar 

  45. Liptser, R.S. and Shiryaev, A.N. (1968): Nonlinear filtering of diffusion type Markov processes. Tr. Mat. Inst. Steklova, 104, 135–80

    Google Scholar 

  46. Rozovskii, B.L. (1972): Stochastic partial differential equations arising in nonlinear filtering problems. Usp. Mat. Nauk SSSR, 27, 3, 213–4.

    MathSciNet  MATH  Google Scholar 

Note and References.2

  1. Liptser, R.S. and Shiryaev, A.N. (1989): Theory of Martingales. Kluwer, Dordrecht (Russian edition 1986 )

    Book  MATH  Google Scholar 

  2. Liptser, R. and Muzhikanov, P. (1998): Filtering with a limiter. J. Appl. Math. Stochastic Anal., 11, 3, 289–300

    Article  MathSciNet  MATH  Google Scholar 

  3. Lototsky, S.V., Mikulevicius, R. and Rozovskii, B.L. (1997): Nonlinear filtering revisited. A spectral approach. SIAM J. Control Optimization, 35, 2, 435–61

    Article  MathSciNet  MATH  Google Scholar 

  4. Lototsky, S.V., Rao, C. and Rozovskii, B.L. (1996): Fast nonlinear filter for continuous-discrete time multiple models. Proc. 35th IEEE Conf. on Decision and Control ( Kobe, Japan ), 4060–4

    Google Scholar 

  5. Lototsky, S.V. and Rozovskii, B.L. (1996): Recursive multiple Wiener integral expansion for nonlinear filtering of diffusion processes. In: A Festschrift in Honor of M.M. Rao, Marcel Dekker

    Google Scholar 

  6. Lototsky, S.V. and Rozovskii, B.L. (1996): Nonlinear filtering revisited: separation of parameters and observations, II. Proc. 35th IEEE Conf. on Decision and Control, ( Kobe, Japan )

    Google Scholar 

  7. Lototsky, S.V. and Rozovskii, B.L. (1998): Recursive nonlinear filter for a continuous-discrete time model. IEEE Trans. Autom. Control, 43, 8, 1154–8

    Article  MathSciNet  MATH  Google Scholar 

  8. Mikulevicius, R. and Rozovskii, B.L. (1993): Separation of observations and parameters in nonlinear filtering. Proc. 32nd IEEE Conf. on Decision and Control, vol 2. IEEE Control Systems Society, 1559–64

    Google Scholar 

  9. Mikulevicius, R. and Rosovskii,B.L. (1999): Martingale problems for stochastic PDE’s. In: Stochastic partial differential equations: six perspectives, Carmona, Rene A. et al. (eds), American Mathematical Society. Math. Surv. Monogr. 64, 243–325

    Google Scholar 

  10. Rozovskii, B.L. (1990): Stochastic Evolution Systems. Linear Theory and Application to Nonlinear Filtering. Reidel, Dordrecht

    Google Scholar 

  11. Krylov, N.V. and Rozovskii, B.L. (1977): On the Cauchy problem for linear stochastic partial differential equations. Mat. SSSR Izv., 11, 1267–84

    Google Scholar 

  12. Krylov, N.V. and Rozovskii, B.L. (1981): Stochastic evolution equations. J. Soy. Math., 16, 1233–76

    Google Scholar 

  13. Kushner, H.J. (1964): On the differential equations satisfied by conditional probability densities of Markov processes. SIAM J. Control Optimization, 2, 106–19

    MathSciNet  MATH  Google Scholar 

  14. Kushner, H.J. (1967): Dynamical equations for nonlinear filtering. J. Differ. Equations, 3, 179–90

    Article  MathSciNet  MATH  Google Scholar 

  15. Liptser, R.S., Steinberg, Y., Bobrovski, B.Z. and Schuss, Z. (1996): Fixed lag smoothing of scalar diffusions. Part I: the filtering-smoothing equation. Stochastic Processes Appl., 64, 237–55

    Article  MATH  Google Scholar 

  16. Zakai, M. (1969): On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Gebiete, 11, 230–33

    Article  MathSciNet  MATH  Google Scholar 

  17. Baras, J.S. (1991): Real-time architecture for the Zakai equations and applications. In: Stochastic Analysis. E. Mayer-Wolf et al. (eds), Academic, Boston, MA

    Google Scholar 

  18. Bensoussan, A., Glowinski, R. and Rascanu, A. (1990): Approximation of the Zakai equation by the splitting-up method. SIAM J. Control Optimization, 28, 1420–31

    Article  MathSciNet  MATH  Google Scholar 

  19. Budhiraja, A. and Kallianpur, G. (1996): Approximations to the solution of the Zakai equation using multiple Wiener and Stratonovich integral expansions. Stochastics Stochastics Rep., 56, 3–4, 271–315

    Google Scholar 

  20. Elliott, R.J., Aggoun, L. and Moore, J.B. (1995): Hidden Markov Models. Springer-Verlag, New York Berl in Heidelberg

    MATH  Google Scholar 

  21. Florchinger, P. and Le Gland, F. (1990): Time-quantization of the Zakai equation for diffusion processes observed in correlated noise. In: 9th Conference on Analysis and Optimization of Systems, A. Bensoussan and J.L. Lions (eds), Lecture Notes in Control and Information Sciences 144, Springer-Verlag, Berlin Heidelberg New York

    Google Scholar 

  22. Itô, K. (1996): Approximation of Zakai equation for nonlinear filtering. SIAM J. Control Optimization, 34, 620–34

    Article  MATH  Google Scholar 

  23. Rozovskii, B.L. (1991): A simple proof of uniqueness for Kushner and Zakai equations. In: Stochastic Analysis. E. Mayer-Wolf et al. (eds). Academic, New York, 449–58

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering Systems, Tel Aviv University, Ramat Aviv, P.O. Box 39040, 69978, Tel Aviv, Israel

    Robert S. Liptser

  2. Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, 117966, Moscow, Russia

    Albert N. Shiryaev

Authors
  1. Robert S. Liptser
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  2. Albert N. Shiryaev
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Liptser, R.S., Shiryaev, A.N. (2001). General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes. In: Statistics of Random Processes. Applications of Mathematics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-13043-8_9

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  • DOI: https://doi.org/10.1007/978-3-662-13043-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08366-2

  • Online ISBN: 978-3-662-13043-8

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