${\widetilde}{\Lambda}_{c}^{-}\in\cP(|{\widetilde}{\mathbb{E}}^{+}|,|{\widetilde}{\mathbf{E}}^{-}|)$, ${\widetilde}{\mG}_{c}^{+}\in\cQ(|{\widetilde}{\mathbf{E}}^{+}|)$, and ${\widetilde}{\mG}_{c}^{-}\in\cQ(|{\widetilde}{\mathbf{E}}^{-}|)$, such that $$\begin{aligned}
    \label{eq:WHZ}
        {\widetilde}{V}^{-1}\left({\widetilde}{\mG}-c\,{\widetilde}{\mI}\right)\left(\begin{array}{cc} {\widetilde}{\mI}^{+} & {\widetilde}{\Lambda}_{c}^{-} \\ {\widetilde}{\Lambda}_{c}^{+} & {\widetilde}{\mI}^{-} \end{array}\right)=\left(\begin{array}{cc} {\widetilde}{\mI}^{+} & {\widetilde}{\Lambda}_{c}^{-} \\ {\widetilde}{\Lambda}_{c}^{+} & {\widetilde}{\mI}^{-} \end{array}\right)\left(\begin{array}{cc} {\widetilde}{\mG}_{c}^{+} & 0 \\ 0 & -{\widetilde}{\mG}_{c}^{-} \end{array}\right);
        \end{aligned}$$

-   the matrices ${\widetilde}{\Lambda}_{c}^{+}$, ${\widetilde}{\Lambda}_{c}^{-}$, ${\widetilde}{\mG}_{c}^{+}$, and ${\widetilde}{\mG}_{c}^{-}$, admit the following probabilistic representations, $$\begin{aligned}
    \label{eq:TildeTau0+}
        {\widetilde}{\Lambda}_{c}^{+}((k,i),(\ell,j))&={\widetilde}{\bE}\left(\left.e^{-c{\widetilde}{\tau}_{0}^{+}}\mathbf{1}_{\{Z_{{\widetilde}{\tau}_{0}^{+}}=(\ell,j)\}}\,\right|Z_{0}=(k,i)\right),\quad (k,i)\in{\widetilde}{\mathbf{E}}^{-},\,(\ell,j)\in{\widetilde}{\mathbf{E}}^{+},\\
        \label{eq:TildeTau0-} {\widetilde}{\Lambda}_{c}^{-}((k,i),(\ell,j))&={\widetilde}{\bE}\left(\left.e^{-c{\widetilde}{\tau}_{0}^{-}}\mathbf{1}_{\{Z_{{\widetilde}{\tau}_{0}^{-}}=(\ell,j)\}}\,\right|Z_{0}=(k,i)\right),\quad (k,i)\in{\widetilde}{\mathbf{E}}^{+},\,(\ell,j)\in{\widetilde}{\mathbf{E}}^{-},\\
        \label{eq:TildeTaut+}
        e^{t\,{\widetilde}{\mG}_{c}^{+}}((k,i),(\ell,j))&={\widetilde}{\bE}\left(\left.e^{-c{\widetilde}{\tau}_{t}^{+}}\mathbf{1}_{\{Z_{{\widetilde}{\tau}_{t}^{+}}=(\ell,j)\}}\,\right|Z_{0}=(k,i)\right),\quad (k,i)\in{\widetilde}{\mathbf{E}}^{+},\,(\ell,j)\in{\widetilde}{\mathbf{E}}^{+},\\
        \label{eq:TildeTaut-}
        e^{t\,{\widetilde}{\mG}_{c}^{-}}((k,i),(\ell,j))&={\widetilde}{\bE}\left(\left.e^{-c{\widetilde}{\tau}_{t}^{-}}\mathbf{1}_{\{Z_{{\widetilde}{\tau}_{t}^{-}}=(\ell,j)\}}\,\right|Z_{0}=(k,i)\right),\quad (k,i)\in{\widetilde}{\mathbf{E}}^{-},\,(\ell,j)\in{\widetilde}{\mathbf{E}}^{-},
        \end{aligned}$$ for any $t\geq 0$.

In what follows we will use the “+” part of the above formulae and only for $k=0$. Accordingly, we define (recall ) $$\begin{aligned}
\label{eq:TildeTau0+0}
{\widetilde}{\Pi}_{c}^{+}(i,j,\ell)&:={\widetilde}{\Lambda}_{c}^{+}((0,i),(\ell,j))={\widetilde}{\bE}^{i}\left(e^{-c{\widetilde}{\tau}_{0}^{+}}\mathbf{1}_{\{Z_{{\widetilde}{\tau}_{0}^{+}}=(\ell,j)\}}\right),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{+},\,\ell\in\bN,\\
\label{eq:TildeTaut+0}
{\widetilde}{\Psi}_{c}^{+}(t,i,j,\ell)&:=e^{t\,{\widetilde}{\mG}_{c}^{+}}((0,i),(\ell,j))={\widetilde}{\bE}^{i}\left(e^{-c{\widetilde}{\tau}_{t}^{+}}\mathbf{1}_{\{Z_{{\widetilde}{\tau}_{t}^{+}}=(\ell,j)\}}\right),\quad i,j\in\mathbf{E}^{+},\,\ell\in\bN,\,t\geq 0.\end{aligned}$$ Note that, for any $t\geq 0$, ${\widetilde}{v}(Z_{t})=v(Y_{t})$, which implies that ${\widetilde}{\varphi}_{t}=\psi_{t}$, and so $\rho_{t}^{+}={\widetilde}{\tau}_{t}^{+}$, $\rho_{t}^{-}={\widetilde}{\tau}_{t}^{-}$. Hence, by taking summations over all $\ell\in\bN$ in and , we obtain that $$\begin{aligned}
\label{eq:SumTildeTau0+}
{\widetilde}{\bE}^{i}\left(e^{-c\rho_{0}^{+}}{\mathbbm{1}}_{\{Y_{\rho_{0}^{+}}=j\}}\right)&=\sum_{\ell=0}^{n}\,{\widetilde}{\Pi}_{c}^{+}(i,j,\ell),\quad i\in\mathbf{E}^{-},\,j\in\mathbf{E}^{+},\\
\label{eq:SumTildeTaut+}
{\widetilde}{\bE}^{i}\left(e^{-c\rho_{t}^{+}}{\mathbbm{1}}_{\{Y_{\rho_{t}^{+}}=j\}}\right)&=\sum_{\ell=0}^{n}\,{\widetilde}{\Psi}_{c}^{+}(t,i,j,\ell),\quad i,j\in\mathbf{E}^{+},\,t\geq 0.\end{aligned}$$

Observe that, in view of , if $U:{\widetilde}{\Omega}\rightarrow\bR$ is an ${\widetilde}{\cF}_{\infty}^{Y}$-measurable bounded random variable, then for any $i\in\mathbf{E}$, $$\begin{aligned}
{\widetilde}{\bE}^{i}(U)=\int_{0}^{\infty}\int_{s_{1}}^{\infty}\cdots\int_{s_{n-1}}^{\infty}\overline{\bE}^{i}(U;s_{1},\ldots,s_{n})\prod_{k=1}^{n}\left(q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})}\right)ds_{n}\cdots ds_{2}ds_{1}.\end{aligned}$$ Therefore, in light of Corollary \[thm:XYWHRelation\], and , we have that $$\begin{aligned}
 \widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)&:=\sum_{\ell=0}^{n}\,{\widetilde}{\Pi}_{c}^{+}(i,j,\ell)\\
&=\int_{0}^{\infty}\int_{s_{1}}^{\infty}\ldots\int_{s_{n-1}}^{\infty}\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})\prod_{k=1}^{n}\left(q_{k}e^{-q_{k}(s_{k}-s_{k-1})}\right)ds_{n}\cdots ds_{2}\,ds_{1}.\end{aligned}$$ $$\begin{aligned}
 \widehat{\Psi}_{c}^{+}(t,i,j;q_1,\ldots,q_n)&:=\sum_{\ell=0}^{n}\,{\widetilde}{\Psi}_{c}^{+}(t,i,j,\ell)\\
&=\int_{0}^{\infty}\!\!\!\int_{s_{1}}^{\infty}\!\!\ldots\!\int_{s_{n-1}}^{\infty}\!\Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})\prod_{k=1}^{n}\left(q_{k}e^{-q_{k}(s_{k}-s_{k-1})}\right)ds_{n}\!\cdots ds_{2}\,ds_{1}.\end{aligned}$$ By change of variables, we obtain $$\begin{aligned}
\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)
&=\int_{0}^{\infty}\int_{0}^{\infty}\ldots\int_{0}^{\infty}\Pi_{c}^{+}(i,j;t_{1},\ldots,t_{1}+\ldots+t_{n})\prod_{k=1}^{n}\left(q_{k}e^{-q_{k}t_{k}}\right)dt_{1}\cdots dt_{n},\\
\widehat{\Psi}_{c}^{+}(i,j;q_1,\ldots,q_n)
&=\int_{0}^{\infty}\int_{0}^{\infty}\ldots\int_{0}^{\infty}\Psi_{c}^{+}(i,j;t_{1},\ldots,t_{1}+\ldots+t_{n})\prod_{k=1}^{n}\left(q_{k}e^{-q_{k}t_{k}}\right)dt_{1}\cdots dt_{n}.\end{aligned}$$ The above two equalities together with the argument in Section \[sec:inverLaplaceTransform\], implies that $$\begin{aligned}
q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n), 