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Estimates for perturbations of average Markov decision processes with a minimal state and upper bounded by stochastically ordered Markov chains. (English) Zbl 1249.90313
Summary: This paper deals with Markov decision processes (MDPs) with real state space which attain their minimum, and which are upper-bounded by (uncontrolled) stochastically ordered (SO) Markov chains. We consider MDPs with (possibly) unbounded costs, and to evaluate the quality of each policy, we use the objective function known as the average cost. For this objective function, we consider two Markov control models \(\mathbb {P}\) and \(\mathbb {P}_1\). \(\mathbb {P}\) and \(\mathbb {P}_1\) have the same components except for the transition laws. The transition \(q\) of \(\mathbb {P}\) is taken as unknown, and the transition \(q_1\) of \(\mathbb {P}_1\), as a known approximation of \(q\). Under certain irreducibility, recurrence and ergodic conditions imposed on the bounding SO Markov chain (these conditions give the rate of convergence of the transition probability to the invariant measure in \(t\)-steps, \(t=1,2,\ldots \)), the difference between the optimal cost to drive \(\mathbb {P}\) and the cost obtained to drive \(\mathbb {P}\) using the optimal policy of \(\mathbb {P}_1\) is estimated. This difference is defined as the index of perturbations and upper bounds of it are provided. An example to illustrate the theory developed here is added.

90C40 Markov and semi-Markov decision processes
93E20 Optimal stochastic control
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