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A characterization of the optimal risk-sensitive average cost in finite controlled Markov chains. (English) Zbl 1076.93045
The paper deals with optimal control problems for Markov chains with finite state and action spaces when the performance index is given by the long-run risk-sensitive average cost criterion. It is assumed that the transition law of the controlled chain satisfies the simultaneous Doeblin condition.
The main result characterizes for any positive risk-sensitivity coefficient the optimal value function as the infimum of a family of functions on the state space. It is then shown that for large values of the risk-sensitivity coefficient the optimal cost is not necessarily constant and the optimality equation may have no solution at all.

MSC:
93E20 Optimal stochastic control
60F10 Large deviations
93C55 Discrete-time control/observation systems
60J05 Discrete-time Markov processes on general state spaces
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