Value iteration for controlled Markov chains with risk sensitive cost criterion

The paper studies the value iteration algorithm for risk sensitive controlled Markov chains. For risk neutral (average cost) Markov decision processes, this algorithm is a standard technique to obtain approximations to a solution of the dynamic programming equation (O. Hernandez-Lerma, 1989; R. Cavazos-Cadena, 1997). We define the risk sensitive control problem of discrete time controlled Markov processes on an infinite horizon, and the first problem is to find suitable conditions under which there exists a solution to the dynamic programming equation when the control set is a compact metric space. We approach this problem, defining the dynamic programming operator (G.B. Di Masi and L. Stettner, 1997). Using the Banach fixed point theorem, it can be proved that this operator has a span-fixed point. The second basic problem is to proved that the value iteration algorithm can be implemented. This is solved using the contractive properties of the operator T.