Particular solutions of the Chapman—Kolmogorov equation for multi-dimensional-state Markov process with continuous time

Abstract

The bilinear integral Chapman—Kolmogorov equation defines the dynamics behavior of a Markov process. The task of its immediate solution without the linearization was set in 1932 by S. N. Bernstein and partially was solved in 1961 by O. V. Sarmanov as bilinear series. In 2007–2010 author found several partial solutions of the above equation in the form of both a series of the Sarmanov-type and an integral. It was assumed that the state space of a Markov process was one-dimensional. In the article three particular solutions are found as integrals for multi-dimensional-state Markov process. Results are illustrated with five examples, one of which shows that it is the solution of the original equation that does not have a probabilistic sense. Refs 8.