The Stochastic Equation of the Optimal Filter (Part II)

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


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.


Stochastic Differential Equation Conditional Expectation Wiener Process Hausdorff Space Conditional Density 
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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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