On extremal solutions of controlled nonlinear filtering equations

Abstract

Controlled nonlinear filtering equations with "relaxed controls" arising out of the separated control problem for partially observed diffusions are considered. An equivalence relation is defined on the attainable laws of the joint state and control process by identifying any two of the latter when their one-dimensional marginals agree almost everywhere. Extreme points of the set of such equivalence classes (for fixed initial laws) are shown to correspond to Markov processes.