Turkish Journal of Mathematics




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.


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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