Parameter-Dependent Poisson Equations: Tools for Stochastic Approximation in a Markovian Framework*

The objective of the paper is to revisit a key mathematical technology within the theory of stochastic approximation in a Markovian framework, elaborated in detail in Benveniste et al. (1990): the existence, uniqueness and Lipschitz-continuity of the solutions of a parameter-dependent Poisson equation. The starting point of our investigation is a relatively new, elegant stability theory for Markov processes developed by Hairer and Mattingly (2011). The paper provides a transparent analysis of parameter-dependent Poisson equations with convenient conditions. The application of our results for the ODE analysis of stochastic approximation in a Markovian framework is the subject of a forthcoming paper.