On optimization and convergence of observation channels and quantizers in stochastic control

This paper studies the optimization of observation channels and quantizers in partially observed stochastic control problems. Continuity properties of the optimal cost in channels and quantizers are explored. Sufficient conditions for sequential compactness under total variation and setwise convergence are presented. Conditions for existence of optimal channels and quantizers are established. It is shown that quantizers are extreme points of the space of channels in an appropriate sense.