On the construction problem for uni-instantaneous bilateral birth-death processes (Q862148)

A matrix \(Q = (q_{ij} : i, j \in {\mathbb Z}), q_{ij} \in {\mathbb R}\), is called a pre-generator if \(-\infty\leq q_{ii}\leq 0 \leq q_{ij} <\infty , i \neq j\), and \(\sum_{ j\neq i} q_{ij} \leq - q_{ii}\). The existence conditions for the corresponding uni-instantaneous bilateral birth-death \(Q\)- processes and the problem of construction of the processes are investigated. It is noted that the conclusions for the uni-instantaneous bilateral case can not be directly derived from the known results for the unilateral cases.