Inhomogeneous Semi-Markov Processes, Select Actuarial Tables, and Duration-Dependence in Demography

Abstract

0. Summary This paper starts by giving a number of demographic and actuarial examples of time-inhomogeneous semi-Markovian models. The examples are presented in a uniform terminology, viz. that of forces of transition between states in a system (state space). The states correspond to demographic or actuarial statuses, and the central features of the substantive models are reflected in the pattern of the state space and in the specification of the forces of transition. A sample path usually corresponds to the history of an individual. Jumps between states correspond to demographic events. Forces of transition have been highly useful in a number of fields of application, yet the standard literature on semi-Markov (and the related Markov renewal) processes has found little room for them. The place of these functions in the now classical, time-homogeneous theory is pointed out briefly, and they are used as a connecting link between this theory and that of the corresponding inhomogeneous processes. The rudiments of the latter theory are then spelled out. Its basic notions are introduced; it is shown how the device of an operational time can be used to transform an inhomogeneous semi-Markov process into a homogeneous one (necessary and sufficient conditions are given and a uniqueness theorem is proved); and certain important problems connected with retrospective investigations are pointed out.