Table of Contents Author Guidelines Submit a Manuscript
Journal of Probability and Statistics
Volume 2011, Article ID 259091, 15 pages
http://dx.doi.org/10.1155/2011/259091
Research Article

A Bayes Formula for Nonlinear Filtering with Gaussian and Cox Noise

1Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA
2Department of Mathematics, University of Munich, Theresienstrasse 39, 80333 Munich, Germany
3Center of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

Received 27 May 2011; Accepted 6 September 2011

Academic Editor: L. A. Shepp

Copyright © 2011 Vidyadhar Mandrekar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Kallianpur and C. Striebel, “Estimation of stochastic systems: arbitrary system process with additive white noise observation errors,” Annals of Mathematical Statistics, vol. 39, pp. 785–801, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. G. Kallianpur, Stochastic Filtering Theory, Springer, New York, NY, USA, 1980.
  3. M. Zakai, “On the optimal filtering of diffusion processes,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 11, no. 3, pp. 230–243, 1969. View at Google Scholar · View at Zentralblatt MATH
  4. P. Mandal and V. Mandrekar, “A Bayes formula for Gaussian noise processes and its applications,” SIAM Journal on Control and Optimization, vol. 39, no. 3, pp. 852–871, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. T. Meyer-Brandis and F. Proske, “Explicit solution of a non-linear filtering problem for Lévy processes with application to finance,” Applied Mathematics and Optimization, vol. 50, no. 2, pp. 119–134, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  6. D. Poklukar, “Nonlinear filtering for jump-diffusions,” Journal of Computational and Applied Mathematics, vol. 197, no. 2, pp. 558–567, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. Grigelionis, “Stochastic non-linear filtering equations and semimartingales,” in Nonlinear Filtering and Stochastic Control, S. K. Mitter and A. Moro, Eds., vol. 972 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1982. View at Google Scholar
  8. B. Grigelionis, “Stochastic evolution equations and densities of the conditional distributions,” in Theory and Application of Random Fields, vol. 49 of Lecture Notes in Control and Information Sciences, pp. 49–88, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. N. Aronszajn, “Theory of reproducing kernels,” Transactions of the American Mathematical Society, vol. 68, pp. 337–404, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. E. Parzen, “Regression analysis of continuous parameter time series,” in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 469–489, University of California Press, 1961.
  11. E. Parzen, “Statistical inference on time series by Hilbert space methods,” in Time Series Analysis Papers, pp. 251–382, Holden-Day, London, UK, 1967. View at Google Scholar
  12. J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288, Springer, Berlin, Germany, 2nd edition, 2002.
  13. R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes, vol. 1, Springer, New York, NY, USA, 1977.
  14. T. Meyer-Brandis, “Stochastic Feynman-Kac equations associated to Lévy-Itô diffusions,” Stochastic Analysis and Applications, vol. 25, no. 5, pp. 913–932, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH