The Stochastic Equation of the Optimal Filter (Part II)

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

Abstract

In Theorems 8.4.3 and 8.4.4 of Section 8.4 the conditional expectation E t f(X t ) was shown to satisfy a stochastic differential equation for all f in D(Ã) for which the condition \(\int_O^T {E\left| {f\left( {X_t } \right)h_t } \right|^2 < \infty } \) is fulfilled. However, the filtering problem can be regarded as completely solved if we can derive from (8.4.22) a stochastic differential equation for the conditional probability distribution—or the condition probability density—of X t given, ℱt z and if, furthermore, it can be established that the equation has a unique solution. This was achieved in Chapter 10 for the linear theory, and we saw that the general equations of Chapter 8 yielded the Kalman filter. The complete solution of the optimal nonlinear filtering problem presents a much more difficult task.

Keywords

Stochastic Differential Equation Conditional Expectation Wiener Process Hausdorff Space Conditional Density 
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.

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Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

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

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