Constructions involving interaction of Markov process's with a random environment

Recently Belopolskaya and Suhov studied Markov process's in a random environment, where the environment changes in likeness to a Markov process. Constructions were made to allow the process to "interact with the environment", this was done in such a manner that the combined process could be studied by studying the individual ones. A new construction is made which has similar properties. This construction can be carried out with more relaxed conditions than the ones previously done.