Joint densities of hitting times for finite state Markov processes

Bielecki T. R., Jeanblanc M., Sezer A. D.

TURKISH JOURNAL OF MATHEMATICS, vol.42, no.2, pp.586-608, 2018 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 42 Issue: 2
  • Publication Date: 2018
  • Doi Number: 10.3906/mat-1608-29
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.586-608
  • Middle East Technical University Affiliated: No


For a finite state Markov process X and a finite collection {Gammak, k is an element of K} of subsets of its state space, let tauk be the first time the process visits the set Gammak. In general, X may enter some of the Gammak at the same time and therefore the vector tau := (tauk, k is an element of K) may put nonzero mass over lower dimensional regions of R+ vertical bar K vertical bar;these regions are of the form Rs = {t R +vertical bar K vertical bar : ti = tj, i, j is an element of s(1) } boolean AND boolean AND l=2& s vertical bar {t:tm < ti = tj, i,j is an element of s(l), m is an element of s(i - 1) } where s is any ordered partition of the set K and s(j) denotes the jth subset of K in the partition s. When vertical bar s vertical bar < vertical bar K vertical bar, the density of the law of tau over these regions is said to be "singular" because it is with respect to the 181-dimensional Lebesgue measure over the region Rs. We derive explicit/recursive and simple to compute formulas for these singular densities and their corresponding tail probabilities over all Rs as s ranges over ordered partitions of K. We give a numerical example and indicate the relevance of our results to credit risk modeling.