Abstract
We consider the problem of filtering one-dimensional diffusions with nonlinear drift coefficients, transmitted through a nonlinear fow noise channel. We construct an asymptotic solution to Zakai’s equation for the unnormalized conditional probability density of the signal, given the noisy measurements. This expansion is used to find the asymptotic expansion of the minimum error variance filter and its mean square estimation error (MSEE). We construct approximate filters whose MSEE agrees with that of the optimal one to a given degree of accuracy. The dimension of the approximate filter increases with the required degree of accuracy. Similarly, we expand the maximum a posteriori probability estimator and the minimum energy estimator and compare their performance. We also discuss some extended Kalman filters and present some examples.
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