Estimation of Stochastic Multistage Inclusions in Denumerable Sets

Abstract

Multistage stochastic inclusions are considered in the product of two at most denumerable phase spaces. The state projection to one of them is considered observable, and to another is not observable. The right part of the inclusion is a multimapping depending on the previous state and a random element of some probability space. Random transition on each step does not depend from previous steps. Three ways of estimation of not observed states are considered which are based on different types of forming of the set of transitional probabilities. It is shown that these ways, generally speaking, lead to various sets of conditional distributions for not observed states of the process. In the case of non-atomic probability spaces the theorem of sufficiency for coincidence of the considered schemes of filtration is proved.