Section: New Results

Approximation of average cost Markov decision processes using empirical distributions and concentration inequalities

Participants : Francois Dufour, Tomas Prieto-Rumeau.

We consider a discrete-time Markov decision process with Borel state and action spaces, and possibly unbounded cost function. We assume that the Markov transition kernel is absolutely continuous with respect to some probability measure $\mu$. By replacing this probability measure with its empirical distribution ${\mu }_n$ for a sample of size $n$, we obtain a finite state space control problem, which is used to provide an approximation of the optimal value and an optimal policy of the original control model. We impose Lipschitz continuity properties on the control model and its associated density functions. We measure the accuracy of the approximation of the optimal value and an optimal policy by means of a non-asymptotic concentration inequality based on the 1–Wasserstein distance between $\mu$ and ${\mu }_n$. Obtaining numerically the solution of the approximating control model is discussed and an application to an inventory management problem is presented. This work has been published in Stochastics An International Journal of Probability and Stochastic Processes: [26] .