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Change of measures has been an effective method in stochastic control andanalysis; in continuous-time control this follows Girsanov's theorem applied toboth fully observed and partially observed models, in decentralized stochasticcontrol (or stochastic dynamic team theory) this is known as Witsenhausen'sstatic reduction, and in discrete-time classical stochastic control Borkar hasconsidered this method for partially observed Markov Decision processes(POMDPs) generalizing Fleming and Pardoux's approach in continuous-time. Thismethod allows for equivalent optimal stochastic control or filtering in a newprobability space where the measurements form an independent exogenous processin both discrete-time and continuous-time and the Radon-Nikodym derivative(between the true measure and the reference measure formed via the independentmeasurement process) is pushed to the cost or dynamics. However, for this to beapplicable, an absolute continuity condition is necessary. This raises thefollowing question: can we perturb any discrete-time sequential stochasticcontrol problem by adding some arbitrarily small additive (e.g. Gaussian orotherwise) noise to the measurements to make the system measurements absolutelycontinuous, so that a change-of-measure (or static reduction) can be applicablewith arbitrarily small error in the optimal cost? That is, are all sequentialstochastic (single-agent or decentralized multi-agent) problems $\epsilon$-awayfrom being static reducible as far as optimal cost is concerned, for any$\epsilon > 0$? We show that this is possible when the cost function is boundedand continuous in controllers' actions and the action spaces are convex. Wealso note that the solution and the cost obtained for the perturbed system isrealizable (under a randomized policy) for the original model.

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