On optimality in probability and almost surely for processes with a communication property. I. The discrete time case

We establish conditions under which the strategy minimizing the expected value of a cost functional has a much stronger property; namely, it minimizes the random cost functional itself for all realizations of the controlled process belonging to a set, the probability of which is close to one for large time horizons. The main difference of the conditions mentioned from those obtained earlier is that the former do not deal with value function properties but concern a possibility of transition of the controlled process from one state to another in a time with a finite mean. It makes the verification of these conditions in a number of situations of the general form much easier. The first part of the paper concerns processes in discrete time, and second part will be devoted to processes in continuous time.