Optimal controls for partially observed stochastic systems: an infinitesimal approach

Abstract

The stochastic control system is investigated, where ƒ,g depend on the past (XS) S≤t of the state X t , and the control u t depends on partial observations y =(Y t ) made at a fixed finite sequence of times and lying in a countable observation space. We show ihai for a wide class of functions ƒ, g and terminal cost there is always an optimal relaxed control, and in certain circumstances there is an optimal ordinary control. We work on a hyperfinite Loeb space (a very rich probability space constructed using Robinson's nonstandard analysis), where, with a fixed Brownian motion b we have an explicit solution to the above equation for each control u. An optimal internal (i.e. nonstandard) control is easily obtained. We show that an internal control U corresponds naturally to a standard relaxed control v whose solution process is pathwise the same as that for U.