Sequential Stochastic Control (Single or Multi-Agent) Problems Nearly Admit Change of Measures with Independent Measurements

Abstract

Change of measures has been an effective method in stochastic control and analysis; in continuous-time control this follows Girsanov's theorem applied to both fully observed and partially observed models, in decentralized stochastic control (or stochastic dynamic team theory) this is known as Witsenhausen's static reduction, and in discrete-time classical stochastic control Borkar has considered this method for partially observed Markov Decision processes (POMDPs) generalizing Fleming and Pardoux's approach in continuous-time. This method allows for equivalent optimal stochastic control or filtering in a new probability space where the measurements form an independent exogenous process in both discrete-time and continuous-time and the Radon-Nikodym derivative (between the true measure and the reference measure formed via the independent measurement process) is pushed to the cost or dynamics. However, for this to be applicable, an absolute continuity condition is necessary. This raises the following question: can we perturb any discrete-time sequential stochastic control problem by adding some arbitrarily small additive (e.g. Gaussian or otherwise) noise to the measurements to make the system measurements absolutely continuous, so that a change-of-measure (or static reduction) can be applicable with arbitrarily small error in the optimal cost? That is, are all sequential stochastic (single-agent or decentralized multi-agent) problems $ε$-away from being static reducible as far as optimal cost is concerned, for any $ε> 0$? We show that this is possible when the cost function is bounded and continuous in controllers' actions and the action spaces are convex. We also note that the solution and the cost obtained for the perturbed system is realizable (under a randomized policy) for the original model.