Abstract
We show how it is possible to construct efficient duration dependent semi-Markov reliability models by considering recurrence time processes. We define generalized reliability indexes and we show how it is possible to compute them. Finally, we describe a possible application in the study of credit rating dynamics by considering the credit rating migration as a reliability problem.
Highlights
Homogeneous semi-Markov processes HSMPs were defined by 1, 2
This paper introduces, for the first time to the authors’ knowledge, initial and final backward and forward processes in a continuous time homogeneous semi-Markov environment at the same time
By means of this new approach a generalization of the transition probabilities of an HSMP is given and we show how it is possible to consider the time spent by the system in the starting state and in the final state
Summary
Homogeneous semi-Markov processes HSMPs were defined by 1, 2. Journal of Applied Mathematics and Decision Sciences the transitions happen after random durations in general are not described by memoryless distribution functions exponential or geometric This is the reason why HSMP fits better the Markov one in reliability problems; it offers the possibility of being able to use any distribution function. In order to study the duration dependence we attach the backward and forward recurrence time processes to the HSMP and we consider them simultaneously at the beginning and at the end of the considered time interval. The usefulness of the results is illustrated in the applicative section on the credit risk rating dynamic which is one of the most important problems in financial literature It consists of computing the default probability of a firm going into debt.
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