Stochastic control of discrete Markov process parameter

Consider a functioning of discrete Markov process in random environment. That is the process behavior is defined by some randomly varying parameter. Suppose that stationary distributions of the process under fixed meanings of the parameter are known. A problem is to choose rules of parameter ramdom variation so that the stationary distribution of obtained process equals to probability mixture of stationary distributions of this process under fixed parameter meanings. Traditional product theorem gives a respond to this question only if stationary distributions, which correspond to the fixed parameter meanings, coincide with each other. In this paper new and sufficiently simple condition on random parameter variation is introduced. This condition gives a solution of considered problem in general case.