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

Phase-type distribution has been one of the most important probabilistic tools in the analysis of complex stochastic system evolution. However, in their empirical study, Frydman (2005) and Frydman and Schuermann (2008) found that the inclusion of available credit rating information helps improve Nelson-Aalen estimation of cumulative default intensity on corporate bonds. These empirical findings make it clear that the attribution of constant intensity is not strongly supported by actual data. Motivated by these facts, we develop a doubly stochastic jump Markov process whose intensity is driven by observed explanatory covariates. The approach is based on the characterization of a jump Markov process in terms of a discrete-time Markov chain allowing to jump independently at doubly stochastic Poisson process (developed by Cox (1955), Kingman (1964), Serfozo (1972), and Bremaud (1981)) whose jump intensity is adapted to the currently available information of observed covariates. We adopt the Cox forward intensity approach of Duan et al. (2012) allowing the baseline to depend on state and time. Unlike the stochastic model of Jakubowski and Nieweglowski (2008), (2010), and Bielecki et al. (2015), our approach does not require to specify the exact knowledge of dynamics nor the whole trajectory of covariates. Moreover, it provides a straightforward and direct method for simulating the sample paths and in deriving transition matrix and lifetime distribution of the jump Markov process based on observed covariates. In particular, the transition matrix generalizes the Jarrow-Lando-Turnbull model, see Jarrow et al. (1997), of credit rating transition, whereas the distribution extends earlier works of Neuts (1975), (1981) on the phase-type model. We also propose conditional forward intensity of future occurrences of the jump Markov process. Some numerical examples in credit risk are performed to motivate the results. Numerical outcomes on default intensity exhibit similar types of behavior found in empirical study of Duffie et al. (2007), (2009) and Duan et al. (2012). Given their closed forms and ability to capture observed covariates and heterogeneity, the results should offer promising features for applications