Existence, uniqueness, and constructions for stochastically monotone \(Q\)-processes (Q1964660)

A (standard) transition function \(P(t)=\{P_{ij} (t):i,j \in E\}\) for a continuous-time Markov chain on \(E=\{0,1, \dots\}\) is said to be stochastically monotone if \(\sum^\infty_{j=k} P_{ij}(t)\) is a non-decreasing function of \(i\) for each fixed \(k\in E\) and \(t\geq 0\). The author addresses the questions of existence and uniqueness of stochastically monotone \(Q\)-functions for a given conservative \(q\)-matrix \(Q\). He shows that there is at most one stochastically monotone \(Q\)-function and gives conditions under which such a \(Q\)-function (i) exists and (ii) equals the minimal \(Q\)-function. Also, it is shown how the stochastically monotone \(Q\)-function can be constructed if it is not the minimal \(Q\)-function.