Risk sensitive control of Markov processes in countable state space

In this paper we consider infinite horizon risk-sensitive control of Markov processes with discrete time and denumerable state space. This problem is solved by proving, under suitable conditions, that there exists a bounded solution to the dynamic programming equation. The dynamic programming equation is transformed into an Isaacs equation for a stochastic game, and the vanishing discount method is used to study its solution. In addition, we prove that the existence conditions are also necessary.