Transient probability functions of finite birth–death processes with catastrophes

A representation of the transient probability functions of finite birth–death processes (with or without catastrophes) as a linear combination of exponential functions is derived using a recursive, Cayley–Hamilton approach. This method of solution allows practitioners to solve for these transient probability functions by reducing the problem to three calculations: determining eigenvalues of the QQ-matrix, raising the QQ-matrix to an integer power and solving a system of linear equations. The approach avoids Laplace transforms and permits solution of a particular transition probability function from state ii to jj without determining all such functions.