  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.
