Discrete Time Non-Homogeneous Semi-Markov Reliability Transition Credit Risk Models and the Default Distribution Functions

This paper shows how to apply discrete time non-homogeneous semi-Markov processes (DTNHSMP) with an age index to credit risk. The idea is to consider the credit risk problem as a reliability model indexed by the age and in this way, many semi-Markov results could be adapted to describe credit risk problem. The default state is seen as a “non working state”. As the semi-Markov process is a generalization of Markov process, the presented model can be seen as a more general migration model. In fact, in semi-Markov environment the distribution function (d.f.) of the waiting time before a transition can be of any type and without the strong constraints of the Markov model. Furthermore, some results on the asymptotic behavior of the presented model are given. This permits the construction of the d.f. of the default random variable for each “working state”. An example, constructed manipulating some Standard & Poor’s (S&P) data, is presented.