particular, $\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})$ are Borel measurable with respect to $(s_{1},\ldots,s_{n})$ in $\Delta_{n}$.

Wiener-Hopf Factorization for $Z=(N,Y)$
---------------------------------------

This subsection is devoted to computing the expectations on the right-hand side in and . This will be done by computing the corresponding expectations related to the *time-homogeneous* Markov chain $Z=(N,Y)$. The latter computation will be done using the classical Wiener-Hopf factorization results for finite state time-homogeneous Markov chains, originally derived in [@Barlow1980].

We begin with a restatement of the classical Wiener-Hopf factorization applied to $Z$. Towards this end, we let ${\widetilde}{\mathbf{E}}^{+}:=\bN_{n}\times\mathbf{E}^{+}$ and ${\widetilde}{\mathbf{E}}^{-}:=\bN_{n}\times\mathbf{E}^{-}$, and ${\widetilde}{v}:{\widetilde}{\mathbf{E}}\rightarrow\bR\setminus\{0\}$ be a function on ${\widetilde}{\mathbf{E}}$ such that ${\widetilde}{v}(k,i)=v(i)$, for all $(k,i)\in{\widetilde}{\mathbf{E}}$. Next, we define the additive functional ${\widetilde}{\varphi}$ and the corresponding first passage times as $$\begin{aligned}
{\widetilde}{\varphi}_{t}:=\int_{0}^{t}{\widetilde}{v}(Z_{u})\operatorname{d \!}u
,\quad
{\widetilde}{\tau}_{t}^{\pm}:=\inf\left\{\left.r\geq 0\,\right|\pm{\widetilde}{\varphi}_{r} > t\right\},\quad t\geq 0.\end{aligned}$$ Let ${\widetilde}{V}:=diag\{{\widetilde}{v}(k,i):(k,i)\in{\widetilde}{\mathbf{E}}\}$ (a diagonal matrix). We denote by ${\widetilde}{\mI}^{\pm}$ the identity matrix of dimension $|{\widetilde}{\mathbf{E}}^{\pm}|$. Finally, $\cQ(m)$ will stand for the set of $m\times m$ generator matrices (i.e., matrices with non-negative off-diagonal entries and non-positive row sums), and $\cP(m,\ell)$ will be the set of $m\times\ell$ matrices whose rows are sub-probability vectors.

\[thm:WHZ\] [@Barlow1980 Theorem 1 $\&$ 2]$\,\,$ Fix $c>0$. Then,

-   there exists a unique quadruple of matrices $({\widetilde}{\Lambda}_{c}^{+},{\widetilde}{\Lambda}_{c}^{-},{\widetilde}{\mG}_{c}^{+},{\widetilde}{\mG}_{c}^{-})$, where ${\widetilde}{\Lambda}_{c}^{+}\in\cP(|{\widetilde}{\mathbf{E}}^{-}|,|{\widetilde}{\mathbf{E}}^{+}|)$, 