Asymptotic behaviour of the survival probabilities in an inhomogeneous semi-Markov model for the migration process in credit risk

We start with the stochastic foundation of the general discrete-time Market of defaultable bonds. We prove that the above market is viable, if and only if there exists an equivalent martingale measure, from which we construct the forward probability measure and under which the discounted default free bond price is a martingale. Assuming that the migration process of defaultable bonds evolves as an inhomogeneous semi-Markov process, we study the asymptotic behaviour of survival probabilities. We provide a method of estimating real-world transition probability sequences for the semi-Markov process, and statistics for testing their homogeneity over time.