Journal of Probability and Statistics
Volume 2011 (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

Vidyadhar Mandrekar,1 Thilo Meyer-Brandis,2 and Frank Proske3

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.

Abstract

A Bayes-type formula is derived for the nonlinear filter where the observation contains both general Gaussian noise as well as Cox noise whose jump intensity depends on the signal. This formula extends the well-known Kallianpur-Striebel formula in the classical non-linear filter setting. We also discuss Zakai-type equations for both the unnormalized conditional distribution as well as unnormalized conditional density in case the signal is a Markovian jump diffusion.