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 described. In this paper a further generalization is presented. In fact, a duration-dependent semi-Markov model is presented for continuous time nonhomogeneous processes. This extension is motivated by theoretical reasons as well by the practical need of making an efficient rating migration model available. In the paper the mono-unreducible topological structure for nonhomogeneous continuous time semi-Markov processes (NHCTSMPs) is introduced and analysed. It represents a sufficient condition for the absorption of the process in the down states of the system. Then, it represents the ideal environment in which to treat credit risk models. The paper is organized as follows. Section 2 gives definitions and notations on NHCTSMP and introduces backward and forward processes and their joint distributions together with that of the SMP. Section 3 studies the monounreducible topological structure and derives the distribution function of going into default state. Section 4 presents a credit risk model and a numerical example. 2. Continuous Time Non-Homogeneous Semi-Markov Process Let us consider two sequences of random variables defined on a complete; filtered probability space as the following:(i) ,
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