Conditional multivariate distributions of phase-type for a finite mixture of Markov jump processes given observations of sample path

This paper presents conditional joint probability distributions of first-exit times to absorbing sets of a finite mixture of right-continuous Markov jump processes given observations of its sample path. Each underlying Markov jump process comprising the mixture has its own intensity matrix and is defined on the same finite state space with overlapping absorbing sets. The main distinctive aspect of the mixture is that despite the underlying processes have the Markov property, the mixture itself in general does not. Conditional transition matrix and the joint probability distribution of the first-exit times form non-stationary functions of time with ability to capture path dependence when conditioning on available information (either full or partial) of the sample path. In particular, initial profile of the joint distribution forms a generalized mixture of the multivariate phase-type distributions of Assaf et al. (1984). When the underlying Markov processes have the same intensity matrix, in which case the mixture becomes a simple Markov jump process, the path dependence and non-stationary properties are removed from the transition matrix and the joint distribution, while the initial profile distribution reduces to Assaf et al. (1984). Some explicit and numerical examples are discussed to illustrate the results.