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The objective of the paper is to revisit a key mathematical technology withinthe theory of stochastic approximation in a Markovian framework, elaborated indetail by Benveniste, M\'etivier, and Priouret (1990): the existence,uniqueness and Lipschitz continuity of the solutions of a parameter-dependentPoisson equation associated with a collection of Markov chains on general statespaces. The setup and the methodology of our investigation is based on anelegant stability theory for Markov chains, developed by Hairer and Mattingly(2011). The paper provides a transparent analysis of parameter-dependentPoisson equations with convenient conditions. The validity of the proposedconditions is verified for a class of controlled queues.

Poisson Equations, Lipschitz Continuity and Controlled Queues