We formulate and study a class of stochastic games by applying the concept of positional games to finite state space Markov decision processes with average and discounted reward optimization criteria. We consider Markov decision processes that may be controlled by several actors (players) as follows. The set of states of the system in a Markov process is divided into several disjoint subsets that represent the position sets for the corresponding players. Each player controls the process only in his position set via the feasible actions in the corresponding states. The aim of each player is to determine which action should be taken in each state of his position set in order to maximize his own average or discounted sum of stage rewards. The step rewards in the states with respect to each player are known for an arbitrary feasible action in the corresponding states of the position sets. We consider the infinite horizon stochastic games and assume that players use stationary strategies of a selection of the actions in the states, i.e. each player in his arbitrary position uses the same action for an arbitrary discrete moment of time. For the considered class of games we are seeking for a Nash equilibrium. We show that for an arbitrary stochastic positional game with discounted payoffs there exits a stationary Nash equilibrium in pure strategies and for an arbitrary stochastic positional game with average payoffs there exists a stationary Nash equilibrium in mixed strategies. Based on constructive prove of these results we propose an approach for determining stationary Nash equilibria for the considered class of stochastic positional games. Some of these results can be found in [1-3].