Reduction of the Zakai equation by invariance group techniques

A general procedure, inspired from that used for deterministic partial differential equations, is presented to reduce the Zakai stochastic Pde of filtering on n to a stochastic Pde on a lower-dimensional space m, with m < n. The method is based upon invariance group techniques. We show how the existence of invariant solutions of the Zakai equation is related to geometric properties of the infinitesimal generator of the signal process. An illustration of the method to a two-dimensional tracking problem with bearings-only measurements is presented. With a specific choice of the bearings-dependent output function, we obtain a continuous model for which the Zakai equation has solutions which can be computed from a one-dimensional stochastic Pde instead of a two-dimensional Pde for the general solution.