Uniqueness criteria for continuous-time Markov chains with general transition structures

Abstract

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

On solutions of Kolmogorovʼs equations for nonhomogeneous jump Markov processes

“…6] provided an explicit construction for multiple transition functions satisfying the backward Kolmogorov equation. For countable-state homogeneous Markov processes, Kendall [14], Kendall and Reuter [15], and Reuter [20] gave examples with non-unique solutions to Kolmogorov equations and Reuter [20] provided necessary and sufficient conditions for their uniqueness; see also Anderson [1] and Chen et al [3]. Ye, Guo, and Hernández-Lerma [21] proved the existence of a transition function that is the minimal non-negative solution to both the backward and forward Kolmogorov equations for a countable state problem with measurable Q-functions.…”

Speed of stability for birth-death processes

“…Since the Q-matrix is conservative, by [10; Lemma 6.52 and Theorem 6.61], (D, D max (D)) is a Dirichlet form and is indeed the maximal one. Note that in the conservative case, every Q-process (in particular, the semigroup generated by a Dirichlet form) satisfies the backward Kolmogorov's equation by [10; Theorem 1.15 (1)]. (a) Let (1.3) hold.…”

“…They satisfy first the backward and then also the forward Kolmogorov's equations by [10;Theorem 6.16]. This is impossible since condition (1.3) is the uniqueness criterion for the process satisfying the Kolmogorov's equations simultaneously, due to Karlin and McGregor (1957a (1); 1994, Theorem 12.7.1)). Note that criterion (1.3) is equivalent to the uniqueness for the process satisfying one of the Kolmogorov equations since every symmetric process as well as the minimal one satisfies both of the equations.…”

Criteria for Feller transition functions

“…Then it follows from [3,Theorem 7] that the q-matrix Q is not regular and the process is not honest, which means that j P ij (t) < 1 for some i ∈ E and some t > 0.…”

“…We give finally an example of honest Feller Q -function P (t) even when {γ i } is unbounded. The rates are given by γ 2k = 2k, γ 2k−1 = 0, λ 2k (2k) 3 , λ 2k−1 = 1, μ k = 0, k 1. we see from [3,Theorem 7] that Q is regular and the minimal Q -function P (t) is a unique and honest Q -function. We take x k = 1 k (k 1) and x 0 = 1.…”