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On the Poisson Equation of Parameter-Dependent Markov Chains
The objective of the paper is to revisit a key mathematical technology within
the theory of stochastic approximation in a Markovian framework, elaborated in
much detail by Benveniste, M\'etivier, and Priouret (1990): the existence,
uniqueness and Lipschitz-continuity of the solutions of parameter-dependent
Poisson equations associated with parameter-dependent Markov chains on general
state spaces. The setup and the methodology of our investigation is based on a
new, elegant stability theory for Markov chains, developed by Hairer and
Mattingly (2011)