Hindawi Publishing Corporation Advances in Decision Sciences Volume 2012, Article ID 123635, 12 pages doi:10.1155/2012/123635 Research Article Monounireducible Nonhomogeneous Continuous Time Semi-Markov Processes Applied to Rating Migration Models Guglielmo D’Amico, 1 Jacques Janssen, 2 and Raimondo Manca 3 1 Department of Pharmacy, University “G. d’Annunzio” of Chieti, Via dei Vestini 31, 66013 Chieti, Italy 2 CESIAF, EURIA, University of Bretagne Occidentale, 6 Avenue L. Gorgeu, CS 93837, 29238 Brest Cedex 3, France 3 MEMOTEF Department, University “La Sapienza” of Roma, Via del Castro Laurenziano 9, 00161 Roma, Italy Correspondence should be addressed to Guglielmo D’Amico, g.damico@unich.it Received 3 April 2012; Accepted 11 September 2012 Academic Editor: C. D. Lai Copyright q 2012 Guglielmo D’Amico et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Monounireducible nonhomogeneous semi- Markov processes are defined and investigated. The mono- unireducible topological structure is a sufficient condition that guarantees the absorption of the semi-Markov process in a state of the process. This situation is of fundamental importance in the modelling of credit rating migrations because permits the derivation of the distribution function of the time of default. An application in credit rating modelling is given in order to illustrate the results. 1. Introduction Semi-Markov processes SMPs are a generalization of Markov processes in which the waiting time distributions before the occurrence of a transition are modelled by any kind of distribution function; see 15. This means that, on the contrary of Markov processes, it is possible to use also no memoryless distributions which determine a duration effect. The duration effect affirms that the time the system is in a state influences the system’s transition probability. One way to detect and quantify this effect, in a SMP, is by using backward and forward recurrence time processes associated to the SMP. In 5, 10 general distributions of the transition probabilities of SMP with backward and forward times are investigated for discrete time nonhomogeneous and for continuous time homogeneous processes, respectively. In these papers a credit risk application is also