Nonparametric estimation for non-homogeneous semi-Markov processes : an application to credit risk

We propose procedures for estimating the time-dependent transition matrices for the general class of finite nonhomogeneous continuous-time semi-Markov processes. We prove the existence and uniqueness of solutions for the system of Volterra integral equations defining the transition matrices, therefore showing that these empirical transition probabilities can be estimated from window censored event-history data. An implementation of the method is presented based on nonparametric estimators of the hazard rate functions in the general and separable cases. A Monte Carlo study is performed to assess the small sample behavior of the resulting estimators. We use these new estimators for dealing with a central issue in credit risk. We consider the problem of obtaining estimates of the historical corporate default and rating migration probabilities using a dataset on credit ratings from Standard & Poor's.