A general procedure, inspired from that used for deterministic partial
differential equations, is presented to reduce the Zakai stochastic P
de of filtering on R-n to a stochastic Pde on a lower-dimensional spac
e R-m, with m < n. The method is based upon invariance group technique
s. We show how the existence of invariant solutions of the Zakai equat
ion is related to geometric properties of the infinitesimal generator
of the signal process. An illustration of the method to a two-dimensio
nal tracking problem with bearings-only measurements is presented. Wit
h a specific choice of the bearings-dependent output function, we obta
in a continuous model for which the Zakai equation has solutions which
can be computed from a one-dimensional stochastic Pde instead of a tw
o-dimensional Pde for the general solution. (C) 1998 Elsevier Science
B.V. All rights reserved.