Skip to main content
Log in

Some recent developments in nonlinear filtering theory

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

The paper gives a survey of recent progress in nonlinear filtering theory including a description of the newly developed finitely additive approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Allinger, F. and Mitter, S. K.: Stochastics, 4 (1981), 339–348.

    Google Scholar 

  2. Balakrishnan, A. V.: ‘A white noise version of the Girsanov Formula’, Proc. of the Symposium on Stochastic Differential Equations, K. Itô (ed.), Kyoto, Japan, 1976.

  3. Balakrishnan, A. V.: J. Appl. Math. Optim. 3 1977, 209–225.

    Google Scholar 

  4. Balakrishnan, A. V.: ‘Non-linear white noise theory’, in P. R. Krishnaiah (ed.), Multivariate Analysis V, North Holland, 1980.

  5. Baras, J. S. Blankenship, G. L. and Hopkins, W. E.: IEEE Trans. on Automatic Control, 28 (1983), 203–214.

    Google Scholar 

  6. Baras, J. S. Blankenship, G. L. and Mitter, S. K.: ‘Nonlinear filtering of diffusion processes’, Proc. IFAC Congr., Kyoto, Japan, 1981.

  7. Benes, V. E.: Stochastics, 5 (1981), 65–92.

    Google Scholar 

  8. Besala, P.: J. Diff. Eqn. 33 (1979), 26–38.

    Google Scholar 

  9. Bodanko, W.: Ann. Polon. Math. XVIII (1966), 79–94.

    Google Scholar 

  10. Clark, J. M. C.: ‘Conditions for one to one correspondence between an observation process and its innovation’, Tech. Rept. Center for Computing and Automation, Imperial College, London, 1969.

    Google Scholar 

  11. Clark, J. M. C.: ‘The design of robust approximations to the stochastic differential equations of nonlinear filtering’, in J. K., Skwirzynski (ed.) Communications Systems and Random Process Theory, NATO Advanced Study Institute Series, Alphen aan den Rijn: Sijthoff and Noordhoff, 1978.

    Google Scholar 

  12. Davis, M. H. A.: ‘Pathwise solutions and multiplicative functionals in non-linear filtering’, 18th IEEE Conference on Decision and Control, Fort Lauderdale, Florida, 1979.

  13. Davis, M. H. A.: Z. Wahrsch. verw. Geb. 54 (1980), 125–139.

    Google Scholar 

  14. Kailath, T.: IEEE Trans. Autom. Control, AC- 13 (1968), 646–655.

    Google Scholar 

  15. Fujisaki, M. Kallianpur, G. and Kunita, H.: Osaka J. Math. 1 (1972), 19–40.

    Google Scholar 

  16. Grigelionis, B.: Liet. Matem. Rink. 12 (1972), 37–51 (English translation: Lithuanian Math. Trans. 11).

    Google Scholar 

  17. Gross, L.: Trans. Amer. Math. Soc. 105 (1962), 372–390.

    Google Scholar 

  18. Hazewinkel, M. and Marcus, S. I.: Stochastics, 7 (1982), 29–62.

    Google Scholar 

  19. Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, North Holland, 1981.

  20. Kallianpur, G. and Striebel, C.: Ann. Math. Statist. 39 (1968), 785–801.

    Google Scholar 

  21. Kallianpur, G. and Striebel, C.: ‘Stochastic differential equations occurring in the estimation of continuous parameter stochastic processes’, Theory of Probability and its Applications, Vol. 14, No. 4, 1969.

  22. Kallianpur, G.: Stochastic Filtering Theory, Springer-Verlag, 1980.

  23. Kallianpur, G. and Karandikar, R. L.: ‘A finitely additive white noise approach to nonlinear filtering’, J. Appl. Math. Optim 10 (1983), 159–185.

    Google Scholar 

  24. Kallianpur, G. and Karandikar, R. L.: Measure valued equations for the optimum filter in finitely additive nonlinear filtering theory, Tech. Rept. #24, Center for Stochastic Processes, University of North Carolina, Chapel Hill, 1983.

    Google Scholar 

  25. Kallianpur, G. and Karandikar, R. L.: ‘The nonlinear filtering problem for the unbounded case’, Tech. Rept. # 33, Center for Stochastic Processes, University of North Carolina, Chapel Hill, 1983.

    Google Scholar 

  26. Kallianpur, G. and Karandikar, R. L.: ‘Markov property of the optimal filter in the finitely additive white noise approach to nonlinear filtering’ (under preparation).

  27. Krylov, N. V.: Teor. Verioatnost i Primenen. 24 (1979), 771–780 (in Russian).

    Google Scholar 

  28. Krylov, N. V. and Rozovskii, B. L.: Math USSR Izv., 11 (1977), 1267–1284.

    Google Scholar 

  29. Krylov, N. V. and Rozovskii, B. L.: Izv. Akad. Nauk, SSSR, Math. Series 42, 1978.

  30. Krylov, N. V. and Rozovskii, B. L.: J. Sov. Math. 16 (1981), 1233–1276.

    Google Scholar 

  31. Kunita, H.: J. Mult. Anal., 1 (1971), 365–393.

    Google Scholar 

  32. Kunita, H.: ‘Stochastic partial differential equations connected with nonlinear filtering’, in S. K. Mitter and A. Moro (eds.), Nonlinear Filtering and Stochastic Control, Lecture Notes in Math. 972, Springer-Verlag (1983).

  33. Kushner, H.: J. Diff. Eqn. 3 (1967), 179–190.

    Google Scholar 

  34. Liptser, R. S. and Shiryaev, A. N.: Statistics of Random Processes, Vol. 1, New York, Springer-Verlag, 1977.

    Google Scholar 

  35. Meyer, P. A.: ‘Un cours sur les integrales stochastiques’, Seminaire de Probabilities X, Lecture Notes in Math. 511, Springer-Verlag (1976).

  36. Mortenson, R. E.: ‘Optimal control of continuous-time stochastic systems’, Report no-ERL-66–1, Electronics Research Laboratory, College of Engineering, University of California, Berkeley, 1966.

    Google Scholar 

  37. Pardoux, E.: ‘Backward and forward stochastic partial differential equations associated with a nonlinear filtering problem’, 18th IEEE Conference on Decision and Control, Fort Lauderdale, Florida, December 1979.

  38. Pardoux, E.: Stochastics, 3 (1979), 127–168.

    Google Scholar 

  39. Pardoux, E.: Stochastics, 6, (1982), 193–231.

    Google Scholar 

  40. Rozovskii, B. L.: Math USSR Sbornik, 25, (1975), 295–322 (English translation).

    Google Scholar 

  41. Stroock, D. W. and Varadhan, S. R. S.: Multidimensional Diffusion Processes, Springer-Verlag, New York, 1979.

    Google Scholar 

  42. Szpirglas, J.: Ann. Inst. Henri Poincare, B, XIV (1978), 33–59.

    Google Scholar 

  43. Zakai, M.: Z. Wahrsch. Verw. Geb. 11 (1969), 230–243.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This research is supported by AFOSR Contract No. F49620 82 C 0009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kallianpur, G., Karandikar, R.L. Some recent developments in nonlinear filtering theory. Acta Appl Math 1, 399–434 (1983). https://doi.org/10.1007/BF00120483

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00120483

AMS (MOS) subject classifications (1980)

Key words

Navigation