Investigation of functionals on paths of a semi-Markov process with a finite state set
For a semi-Markov process \(X\) with finite state space, path functionals \(\eta\) and \(\nu\) are introduced; the latter is a control affecting an additive income functional, but not \(X\). Let \(\tau\) denote the first state transition epoch. If \(\eta<\tau\), which event is assumed to depend only on \(X\) to \(\tau\), then \(X\) is terminated at \(\eta\); otherwise, the time origin shifts to \(\tau\) and \(\eta\), \(\nu\) are revised accordingly. Laplace transform and moment results are obtained for the joint distribution of the termination time, \(\eta^*\), and the accumulated income; in addition, an exponential waiting time limit for \(\eta^*\) is given.
scientific article; zbMATH DE number 1283618