On optimality in probability and almost surely for processes with communication property. II. Continuous time

The paper is a continuation of [T. A. Belkina and V. I. Rotar, Theory Probab. Appl., 50 (2006), pp. 16–33] that deals with conditions under which the strategy minimizing the expected value of the cost functional is asymptotically optimal almost surely or in probability. The former means that the strategy mentioned minimizes the random cost functional itself for all realizations of the controlled process from a set, the probability of which is close to one for large time horizons. The definition of asymptotic optimality in probability is similar with natural changes. The main difference between the conditions of this paper and 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 expectation. In the first part we dealt with the discrete time case; in this second part we cover controlled diffusion processes.