** Authors:**
TOMASZ R. BIELECKI, MONIQUE JEANBLANC, ALİ DEVİN SEZER

** Abstract: **
For a finite state Markov process $X$ and a finite collection $\{
\Gamma_k, k \in K \}$ of subsets of its state space, let $\tau_k$
be the first time the process visits the set $\Gamma_k$. In
general, $X$ may enter some of the $\Gamma_k$ at the same time and
therefore the vector $\bm\tau :=(\tau_k, k \in K)$ may put nonzero
mass over lower dimensional regions of ${\mathbb R}_+^{|K|}$;
these regions are of the form $R_s=\{{\bm t} \in {\mathbb R}_+^{|K|}: t_i = t_j, ~~i,j \in
s(1) \} \cap \bigcap_{l=2}^{|s|} \{{\bm t}:t_m < t_i = t_j,~~ i,j
\in s(l), m \in s(l-1) \}$ where $s$ is any ordered partition of
the set $K$ and $s(j)$ denotes the $j^{th}$ subset of $K$ in the
partition $s$. When $|s| < |K|$, the density of the law of
$\bm\tau$ over these regions is said to be ``singular'' because it
is with respect to the $|s|$-dimensional Lebesgue measure over the
region $R_s.$ We derive explicit/recursive and simple to compute
formulas for these singular densities and their corresponding tail
probabilities over all $R_s$ as $s$ ranges over ordered partitions
of $K$. We give a numerical example and indicate the relevance of
our results to credit risk modeling.

** Keywords: **
Finite state Markov processes, simultaneous hitting times,
densities of singular parts,
multiple first hitting times, generalized multivariate phase-type distributions,
credit risk modeling

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