Joint Hitting-Time Densities for Finite State Markov Processes

Abstract: 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.