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

Caré, Algo and Csáji, Balázs Csanád and Gerencsér, Balázs and Gerencsér, László and Rásonyi, Miklós (2019) Parameter-Dependent Poisson Equations : Tools for Stochastic Approximation in a Markovian Framework. In: 2019 IEEE 58th Conference on Decision and Control (CDC). Proceedings of the IEEE Conference on Decision and Control . IEEE, Piscataway (NJ), pp. 2259-2264. ISBN 9781728113982

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Abstract

The objective of the present paper is to revisit a key mathematical technology within the theory of stochastic approximation in a Markovian framework, elaborated in much detail in [2]: the existence, uniqueness and smoothness (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 [5]. The current 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.

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