Joint densities of hitting times for finite state Markov processes

Abstract

For a finite state Markov process X and a finite collection {Γ k , k ∈ K} of subsets of its state space, let τ k be the first time the process visits the set Γ k. In general, X may enter some of the Γ k at the same time and therefore the vector τ := (τ k , k ∈ K) may put nonzero mass over lower dimensional regions of R |K| + ; these regions are of the form R s = {t ∈ R |K| + : t i = t j , i, j ∈ s(1)} ∩ |s| l=2 {t : t m < t i = t j , i, j ∈ s(l), m ∈ s(l − 1)} where s is any ordered partition of the set K and s(j) denotes the j th subset of K in the partition s. When |s| < |K|, the density of the law of τ over these regions is said to be singular because it is with respect to the |s|-dimensional Lebesgue measure over the region R s. We derive explicit/recursive and simple to compute formulas for these singular densities and their corresponding tail probabilities over all R s as s ranges over ordered partitions of K. We give a numerical example and indicate the relevance of our results to credit risk modeling. * The research of T.R.