ASYMPTOTIC ANALYSIS OF THE OPTIMAL FILTERING PROBLEM FOR ONE-DIMENSIONAL DIFFUSIONS MEASURED IN A LOW NOISE CHANNEL. PART I.

The authors consider the problem of filtering a diffusion with nonlinear drift transmitted through a linear low noise channel. A ray method is used for the construction of approximate solutions to Kushner's and M. Zakai's equations for the normalized and unnormalized conditional probability density function of the signal. A systematic expansion of the mean square estimation error is given, the realization of an asymptotic optimal filter is presented and shown to be one-dimensional. Some examples are worked out, and the relation to some recent work of V. E. Benes and O. Hijab is discussed.