Characterization of the Optimal Risk-Sensitive Average Cost in Denumerable Markov Decision Chains
研究了可数状态空间上风险敏感平均成本准则下的马尔可夫决策链，在通信性和Doeblin条件下证明了最优平均成本的存在性，并用Collatz-Wielandt公式刻画了最优值，给出了最优策略的存在条件。
This work is concerned with Markov decision chains on a denumerable state space. The controller has a positive risk-sensitivity coefficient, and the performance of a control policy is measured by a risk-sensitive average cost criterion. Besides standard continuity-compactness conditions, it is assumed that the state process is communicating under any stationary policy, and that the simultaneous Doeblin condition holds. In this context, it is shown that if the cost function is bounded from below, and the superior limit average index is finite at some point, then (i) the optimal superior and inferior limit average value functions coincide and are constant, (ii) the optimal average cost is characterized via an extended version of the Collatz-Wielandt formula in the theory of positive matrices, and (iii) an optimality inequality is established, from which a stationary optimal policy is obtained. Moreover, an explicit example is given to show that, even if the cost function is bounded, the strict inequality may occur in the optimality relation.