Conditional Phase-Type Distribution Under Doubly Stochastic Jump Markov Processes with Observed Covariates

Empirical evidences on corporate bond found e.g. in Frydman (2005) and Frydman and Schuermann (2008) suggest the facts that the attribution of constant intensity in the credit rating dynamics, represented by finite-state jump Markov processes, is not borne out by actual data. On the other hand, it is known facts that continuous-time finite-state jump Markov processes can be represented by means of Poisson process and embedded discrete-time Markov chains, see, e.g., Ch. 7 of Pardoux (2008), Sec. 5.10 of Resnick (2002) and and Jakubowski and Nieweglowski (2010, 2008). Motivated by the above facts, we consider representation of jump Markov processes in terms of doubly stochastic Poisson process (see e.g. Cox (1955), Kingman (1964), Serfozo (1972), and Bremaud (1981)) whose intensity is driven by observed explanatory covariates. However, we impose weaker conditions than that of specified in Kingman (1964) and Jakubowski and Nieweglowski (2010, 2008) in the construction of such jump Markov process in which the conditional rate of jump arrival in the conditional Poisson process follows the landmarking approach. This approach has been recently introduced in event history analysis by van Houwelingen (2007) and van Houwelingen and Putter (2012). By this approach, we do not require the whole trajectory (dynamics) of the observed covariates. Analogous to Cox (1955) and Jakubowski and Nieweglowski (2010), we call such representation as doubly stochastic jump Markov processes. We derive lifetime distributions until its absorption of doubly stochastic finite state absorbing jump Markov process, and propose generalization of the phase-type distribution introduced in Neuts (1981, 1975). Also, we derive conditional forward intensity of future occurrences of the jump Markov process. The new distribution and intensity are given in closed form and have ability to capture explanatory covariates and heterogeneity. The results can be seen as an alternative structural approach to the reduced-form model of credit default discussed in Duffie et al. (2009, 2007), and Duan et al. (2012). Some numerical examples are discussed to motivate the main results.