are not sub-Markovian. That is, the sums of row elements of $\mG_{k}$ are all zero, for any $k=1,\ldots,n+1$. The results of this paper carry over to the sub-Markovian case by the standard augmentation of the state space.

Next, we consider a function $v:\mathbf{E}\rightarrow\bR\setminus\{0\}$ and we put $$\begin{aligned}
\mathbf{E}^{+}:=\left\{\left.i\in\mathbf{E}\,\right|v(i)>0\right\}\quad\text{and}\quad\mathbf{E}^{-}:=\left\{\left.i\in\mathbf{E}\,\right|v(i)<0\right\}.\end{aligned}$$ We also define the additive functional $$\begin{aligned}
\varphi_{t}:=\int_{0}^{t}v(X_{u})\operatorname{d \!}u,\quad t\geq 0,\end{aligned}$$ and the first passage times $$\begin{aligned}
\tau_{t}^{+}:=\inf\left\{\left.r\geq 0\,\right|\varphi_{r}>t\right\}\quad\text{and}\quad\tau_{t}^{-}:=\inf\left\{\left.r\geq 0\,\right|\varphi_{r}<-t\right\}.\end{aligned}$$

The main goal of this paper is to apply the Wiener-Hopf factorization technique, which we work out in Section \[subsec:th\], to compute the following expectations, $$\begin{aligned}
\label{eq:tau0+}
\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})&:=\mathbb{E}\left(e^{-c\tau_{0}^{+}}{\mathbbm{1}}_{\{X_{\tau_{0}^{+}}=j\}}|X_{0}=i\right),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{+},\\
\label{eq:taut+}
\Psi_{c}^{+}(\ell,i,j;s_{1},\ldots,s_{n})&:=\mathbb{E}\left(e^{-c\tau_{\ell}^{+}}{\mathbbm{1}}_{\{X_{\tau_{\ell}^{+}}=j\}}|X_{0}=i\right),\quad i\in\mathbf{E}^{+},\,j\in\mathbf{E}^{+},\,\ell>0, \\
\label{eq:tau0-}
\Pi_{c}^{-}(i,j;s_{1},\ldots,s_{n})&:=\mathbb{E}\left(e^{-c\tau_{0}^{-}}{\mathbbm{1}}_{\{X_{\tau_{0}^{-}}=j\}}|X_{0}=i\right),\quad i\in\mathbf{E}^{+},\,j\in\mathbf{E}^{-},\\
\label{eq:taut-}
\Psi_{c}^{-}(\ell,i,j;s_{1},\ldots,s_{n})&:=\mathbb{E}\left(e^{-c\tau_{\ell}^{-}}{\mathbbm{1}}_{\{X_{\tau_{\ell}^{-}}=j\}}|X_{0}=i\right),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{-},\,\ell>0.\end{aligned}$$ We will focus on the computation of $\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{+}(\ell,i,j;s_{1},\ldots,s_{n})$. By symmetry, analogous results can be obtained for $\Pi_{c}^{-}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{-}(\ell,i,j;s_{1},\ldots,s_{n})$. To simplify the notations, we will frequently write $\Pi_{c}^{+}(i,j)$ and $\Psi_{c}^{+}(\ell,i,j)$ in place of $\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{+}(\ell,i,j;s_{1},\ldots,s_{n})$, respectively.

A randomization method and the Wiener-Hopf factorization {#subsec:th}
========================================================

In this section we construct a [*time-homogeneous*]{} Markov chain associated to $X$, by randomizing the discontinuity times $s_{1},\ldots,s_{n}$ of the generator $\mG$. This key construction will allow us to compute the expectations and using analogous expectations corresponding to this time-homogeneous chain. The latter expectations can 