Joint Hitting-Time Densities for Finite State Markov Processes

For a finite state Markov process and a finite collection ${ Gamma_k, k in K }$ of subsets of its state space, let $tau_k$ be the first time the process visits the set $Gamma_k$. We derive explicit/recursive formulas for the joint density and tail probabilities of the stopping times ${ tau_k, k in K}$. The formulas are natural generalizations of those associated with the jump times of a simple Poisson process. We give a numerical example and indicate the relevance of our results to credit risk modeling.