Joint densities of hitting times for finite state Markov processes

Date

2018-01-01

Author

Bielecki, Tomasz R.
Jeanblanc, Monique
Sezer, Ali Devin

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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> vertical bar K vertical bar;these regions are of the form R<INF>s</INF> = {t R <INF>+</INF>vertical bar K vertical bar : t<INF>i</INF> = t<INF>j</INF>, i, j is an element of s(1) } boolean AND boolean AND <INF>l=2</INF>&</INF> s vertical bar {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 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 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.

Subject Keywords

Finite state Markov processes, Simultaneous hitting times, Densities of singular parts, Multiple first hitting times, Generalized multivariate phase-type distributions, Credit risk modeling

URI

https://hdl.handle.net/11511/30278

Journal

TURKISH JOURNAL OF MATHEMATICS

DOI

https://doi.org/10.3906/mat-1608-29

Collections

Graduate School of Applied Mathematics, Article

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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.