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This paper studies semi-Markov control models with Borel state and control spaces, and unbounded cost functions, under the average cost criterion. Conditions are given for (i) the existence of a solution to the average cost optimality equation, and for (ii) the existence of strong optimal control policies. These conditions are illustrated with a semi-Markov replacement model.

We deal with semi-Markov control processes (SMCPs) on Borel spaces with unbounded cost and mean holding time. Under suitable growth conditions on the cost function and the mean holding time, together with stability properties of the embedded Markov chains, we show the equivalence of several average cost criteria as well as the existence of stationary optimal policies with respect to each of these criteria.

This work deals with a class of discrete-time zero-sum Markov games whose state process $\left\{x_t\right\}$ evolves according to the equation $x_{t+1}=F(x_t,a_t,b_t,{\xi }_t),$ where $a_t$ and $b_t$ represent the actions of player 1 and 2, respectively, and $\left\{{\xi }_t\right\}$ is a sequence of independent and identically distributed random variables with unknown distribution $\theta$. Assuming possibly unbounded payoff, and using the empirical distribution to estimate $\theta$, we introduce approximation schemes for the value of the game as well as for optimal strategies considering both,...