Joint densities of hitting times for finite state Markov processes

2018-01-01
Bielecki, Tomasz R.
Jeanblanc, Monique
Sezer, Ali Devin
For a finite state Markov process X and a finite collection {Gamma<INF>k</INF>, k is an element of K} of subsets of its state space, let tau<INF>k</INF> be the first time the process visits the set Gamma<INF>k</INF>. In general, X may enter some of the Gamma<INF>k</INF> at the same time and therefore the vector tau := (tau<INF>k</INF>, k is an element of K) may put nonzero mass over lower dimensional regions of R<INF>+</INF> <SUP>vertical bar K vertical bar</SUP>;these regions are of the form R<INF>s</INF> = {t R <INF>+</INF><SUP>vertical bar K vertical bar</SUP> : t<INF>i</INF> = t<INF>j</INF>, i, j is an element of s(1) } boolean AND boolean AND <INF>l=2</INF><SUP>&</INF> <SUP>s vertical bar</SUP> {t:t<INF>m</INF> < t<INF>i</INF> = t<INF>j</INF>, 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 j<SUP>th</SUP> 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 R<INF>s</INF>. We derive explicit/recursive and simple to compute formulas for these singular densities and their corresponding tail probabilities over all R<INF>s</INF> as s ranges over ordered partitions of K. We give a numerical example and indicate the relevance of our results to credit risk modeling.
TURKISH JOURNAL OF MATHEMATICS

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Citation Formats
T. R. Bielecki, M. Jeanblanc, and A. D. Sezer, “Joint densities of hitting times for finite state Markov processes,” TURKISH JOURNAL OF MATHEMATICS, pp. 586–608, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/30278.