On the Study of Forward Kolmogorov System and the Corresponding Problems for Inhomogeneous Continuous-Time Markov Chains

Abstract

An inhomogeneous continuous-time Markov chain X(t) with finite or countable state space under some natural additional assumptions is considered. As a consequence, we study a number of problems for the corresponding forward Kolmogorov system, which is the linear system of differential equations with special structure of the matrix A(t). In the countable situation we have an equation in the space of sequences \(l_1\). The important properties of X(t) (such as weak and strong ergodicity, perturbation bounds, truncation bounds) are closely connected with behaviour of the solutions of the forward Kolmogorov system as \(t \rightarrow \infty \). The main problems and some approaches for their solution are discussed in the paper.