Special solutions of the Chapman–Kolmogorov equation for multidimensional-state Markov processes with continuous time

Abstract

The bilinear Chapman–Kolmogorov equation determines the dynamical behavior of Markov processes. The task to solve it directly (i.e., without linearizations) was posed by Bernstein in 1932 and was partially solved by Sarmanov in 1961 (solutions are represented by bilinear series). In 2007–2010, the author found several special solutions (represented both by Sarmanov-type series and by integrals) under the assumption that the state space of the Markov process is one-dimensional. In the presented paper, three special solutions have been found (in the integral form) for the multidimensional- state Markov process. Results have been illustrated using five examples, including an example that shows that the original equation has solutions without a probabilistic interpretation.