Bielecki T. R. , Jeanblanc M., Sezer A. D.
TURKISH JOURNAL OF MATHEMATICS, vol.42, no.2, pp.586608, 2018 (Journal Indexed in SCI)

Publication Type:
Article / Article

Volume:
42
Issue:
2

Publication Date:
2018

Doi Number:
10.3906/mat160829

Title of Journal :
TURKISH JOURNAL OF MATHEMATICS

Page Numbers:
pp.586608
Abstract
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 181dimensional 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.
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