Exotic Properties of Non Homogeneous Markov and Semi-Markov Systems

In this article we study what we chose to call exotic properties of NHMS and NHSMS. The interplay between stochastic theory of NHMS and NHSMS and other branches of probability, stochastic processes and mathematics, we believe is a fascinating one apart from being important. In many cases the information needed for the evolution of a NHMS is a larger set than the history of the multidimensional process NHMS. In our world where an overflow of information exists almost in all problems, it is almost surely that this will be available. Here, we extend the definition of the NHMS in order to accomodate this case. In this respect we arrive at the defnition of the 𝒢-non homogeneous Markov system. We study the problem of change of measure in a 𝒢-non homogeneousMarkov system. It is proved that under certain conditions the NHMS retains the Markov property, while as expected the basic sequences of transition probabilities change and it is established how they do so. We also find the expected population structure of the NHMS under the new measure in close analytic form. We also define the 𝒢-non homogeneous semi-Markov system and we study the problem of change of measure in a 𝒢-non homogeneous semi-Markov system. It is proved that under certain conditions the NHSMS retains the semi-Markov property while as expected the basic sequences of transition probabilities change and it is established how they do so. We prove that if the input process of memberships is a non homogeneous Poisson process, then asymtotically and under certain easily met in practice conditions, the compensated population structure of the 𝒢-NHMS is a martingale. Finally we prove that the space of all random population structures, under easily met in practice conditions, is a Hilbert space.