## Approximate nonlinear filtering by projection on exponential manifolds of densities.(English)Zbl 0930.93074

The nonlinear Kushner-Stratonovich filtering equation for the state and observation equations $dX_{t}=f_{t}(X_{t}) dt+\sigma _{t}(X_{t}) dW_{t}, X_{0},$
$dY_{t}=h_{t}(X_{t}) dt+dV_{t}, Y_{0}=0,$ is projected on the finite-dimensional manifold of the family of probability densities $$p(\cdot ,\theta), \theta \in \Theta \subseteq \mathbb{R}^{m}.$$ It is shown that the parameters $$\theta _{t}$$ of the projection filter density $$p(\cdot ,\theta _{t})$$ satisfy a finite-dimensional SDE the explicit form of which is given in the paper. The most constructive results are obtained for exponential densities. The known concept of assumed density filter (ADF) is compared with the projection filter. Simulation results comparing the projection filter and the optimal filter for the cubic sensor problem are presented.

### MSC:

 93E11 Filtering in stochastic control theory 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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