Stochastic Filtering Theory pp 273-293 | Cite as

# The Stochastic Equation of the Optimal Filter (Part II)

## 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, ℱ_{t} ^{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## Preview

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