Joint densities of hitting times for finite state Markov processes


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

TURKISH JOURNAL OF MATHEMATICS, cilt.42, ss.586-608, 2018 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 42 Konu: 2
  • Basım Tarihi: 2018
  • Doi Numarası: 10.3906/mat-1608-29
  • Dergi Adı: TURKISH JOURNAL OF MATHEMATICS
  • Sayfa Sayıları: ss.586-608

Özet

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.