The General Filtering Problem and the Stochastic Equation of the Optimal Filter (Part I)

  • Gopinath Kallianpur
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 13)


Before discussing the filtering problem, we prove a number of results in preparation for the martingale approach to the stochastic differential equation of the optimal filter which will be derived in the later sections of this chapter. Let us recall that (Ω,A,P) is a complete probability space and ( t ) (t ∈ R + ) is an increasing family of sub σ-fields of A, and that it will be assumed that all P-null sets belong to ℱ0. The following processes are given on Ω: (S t ) called the signal or system process; (Z t ), the observation process; and (B t ), the noise process. All three are related by the model
$$Z_t = S_t + B_t $$


Markov Process Stochastic Differential Equation Innovation Process Wiener Process Stochastic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • Gopinath Kallianpur
    • 1
  1. 1.Department of StatisticsUniversity of North CarolinaChapel HillUSA

Personalised recommendations