Proof. Using the unnumbered equations right above (8), (10), and (18) of [Reference Albrecher, Ivanovs and Zhou5], we have that

(24) \begin{eqnarray}\mathrm{P}_{x}\big(X(e_{\lambda})\in \mathrm{d}y,e_{\lambda}\,{\lt}\,\tau_{0}^{-}\big)&\,=\,&\lambda\big(\mathrm{e}^{-\Phi_{\lambda}y}W_{\lambda}(x)-W_{\lambda}(x-y)\big)\mathrm{d}y,\quad y\in(0,\infty),\\\mathrm{E}_{y}\big(\mathrm{e}^{\theta X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,T_{a}^{+}\big)&\,=\,&Z(y,\theta)-\frac{W(y)}{\theta-\Phi_{\lambda}}\bigg(\psi(\theta)-\lambda\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}\bigg),\quad y\in[0,a],\nonumber\\\mathrm{E}_{x}\big(\mathrm{e}^{-\lambda\tau_{0}^{-}+\theta X(\tau_{0}^{-})}\big)&\,=\,&Z_{\lambda}(x,\theta)-\frac{\lambda-\psi(\theta)}{\Phi_{\lambda}-\theta}W_{\lambda}(x),\nonumber\end{eqnarray}

where $W(x)=W_{\lambda=0}(x)$ and $Z(x,\theta)=Z_{\lambda=0}(x,\theta)$ . By conditioning on the first Poisson observation time, we have

\begin{eqnarray*}&&\mathrm{E}_{x}\big(\mathrm{e}^{\theta X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,T_{a}^{+}\big)\nonumber\\[3pt]&\,=\,&\mathrm{E}_{x}\big(\mathrm{e}^{\theta X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,e_{\lambda}\leq T_{a}^{+}\big)+\mathrm{E}_{x}\big(\mathrm{e}^{\theta X(\tau_{0}^{-})};\;e_{\lambda}\,{\lt}\,\tau_{0}^{-}\,{\lt}\,T_{a}^{+}\big)\nonumber\\[3pt]&\,=\,&\mathrm{E}_{x}\big(\mathrm{e}^{-\lambda \tau_{0}^{-}+ \theta X(\tau_{0}^{-})}\big)+\int_{0}^{a}\mathrm{E}_{y}\big(\mathrm{e}^{\theta X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,T_{a}^{+}\big)\mathrm{P}_{x}\left(X(e_{\lambda})\in \mathrm{d}y,e_{\lambda}\,{\lt}\,\tau_{0}^{-}\right)\nonumber\\[3pt]&\,=\,&Z_{\lambda}(x,\theta)-\frac{\lambda-\psi(\theta)}{\Phi_{\lambda}-\theta}W_{\lambda}(x)\nonumber\\[3pt]&&+\lambda\int_{0}^{a}\bigg(Z(y,\theta)-\frac{W(y)}{\theta-\Phi_{\lambda}}\bigg(\psi(\theta)-\lambda\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}\bigg)\bigg)\left(\mathrm{e}^{-\Phi_{\lambda}y}W_{\lambda}(x)-W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[3pt]&\,=\,&Z_{\lambda}(x,\theta)-\frac{\lambda-\psi(\theta)}{\Phi_{\lambda}-\theta}W_{\lambda}(x)+\lambda W_{\lambda}(x)\int_{0}^{a}\mathrm{e}^{-\Phi_{\lambda}y}Z(y,\theta)\, \mathrm{d}y-\lambda\int_{0}^{a}Z(y,\theta)W_{\lambda}(x-y)\mathrm{d}y\nonumber\\[3pt]&&-\frac{\lambda}{\theta-\Phi_{\lambda}}\bigg(\psi(\theta)-\lambda\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}\bigg)\left(W_{\lambda}(x)\int_{0}^{a}\mathrm{e}^{-\Phi_{\lambda}y}W(y)\mathrm{d}y-\int_{0}^{a}W(y)W_{\lambda}(x-y)\mathrm{d}y\right)\nonumber\\[3pt]&\,=\,&Z_{\lambda}(x,\theta)-\frac{\lambda-\psi(\theta)}{\Phi_{\lambda}-\theta}W_{\lambda}(x)+W_{\lambda}(x)\left(\lambda\frac{\mathrm{e}^{-\Phi_{\lambda}a}Z(a,\theta)-1}{\theta-\Phi_{\lambda}} +\frac{\psi(\theta)\left(1-\mathrm{e}^{-\Phi_{\lambda}a}Z(a,\Phi_{\lambda})\right)}{\theta-\Phi_{\lambda}}\right)\\[3pt]&&-\lambda\int_{0}^{a}Z(y,\theta)W_{\lambda}(x-y)\mathrm{d}y\nonumber\\[3pt]&&-\frac{1}{\theta-\Phi_{\lambda}}\bigg(\psi(\theta)-\lambda\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}\bigg)\left(W_{\lambda}(x)\left(1-\mathrm{e}^{-\Phi_{\lambda}a}Z(a,\Phi_{\lambda})\right)+W^{(0,\lambda)}_{a}(x)-W_{\lambda}(x)\right)\nonumber\\[3pt]&\,=\,&Z_{\lambda}(x,\theta)-\frac{\lambda-\psi(\theta)}{\Phi_{\lambda}-\theta}W_{\lambda}(x)+\lambda W_{\lambda}(x)\frac{\mathrm{e}^{-\Phi_{\lambda}a}Z(a,\theta)-1}{\theta-\Phi_{\lambda}}\nonumber\\[3pt]&&-\lambda\int_{0}^{a}Z(y,\theta)W_{\lambda}(x-y)\mathrm{d}y\nonumber\\[3pt]&&-\frac{1}{\theta-\Phi_{\lambda}}\left(\psi(\theta)-\lambda\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}\right)\big(W^{(0,\lambda)}_{a}(x)-W_{\lambda}(x)\big)\nonumber\\[3pt]&&+\frac{\lambda}{\theta-\Phi_{\lambda}}\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}W_{\lambda}(x)\left(1-\mathrm{e}^{-\Phi_{\lambda}a}Z(a,\Phi_{\lambda})\right) \\\ &\,=\,&Z_{\lambda}(x,\theta)-\frac{\lambda-\psi(\theta)}{\Phi_{\lambda}-\theta}W_{\lambda}(x)-\frac{1}{\theta-\Phi_{\lambda}}\left(\psi(\theta)-\lambda\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}\right)\left(W^{(0,\lambda)}_{a}(x)-W_{\lambda}(x)\right)\nonumber\\&&-\lambda\int_{0}^{a}Z(y,\theta)W_{\lambda}(x-y)\mathrm{d}y+\frac{\lambda}{\theta-\Phi_{\lambda}}W_{\lambda}(x)\left(\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}-1\right)\nonumber\\&\,=\,&Z_{a}^{(0,\lambda)}(x,\theta)+\frac{W_{a}^{(0,\lambda)}(x)}{\theta-\Phi_{\lambda}}\left(\lambda\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}-\psi(\theta)\right)\!,\nonumber\end{eqnarray*}

which is (22). By (1)–(3), $Z_{a}^{(0,\lambda)}(a,\theta)=Z(a,\theta)$ and $W_{a}^{(0,\lambda)}(a)=W(a)$ . Hence, letting $x\downarrow a$ in (22) yields (23). This completes the proof.

Proof. From (17) of [Reference Albrecher, Ivanovs and Zhou5], it follows that

(26) \begin{eqnarray} \varphi(x;\;\theta,a)=\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{0}^{-}\big) =\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{W(x)}{Z(a,\Phi_{\lambda})},\quad x\leq a.\end{eqnarray}

By conditioning on the first Poisson observation time and using (24) and (26), we have

\begin{eqnarray}\varphi(x;\;\theta,a)&\,=\,&\int_{a}^{\infty}\mathrm{e}^{-\theta (y-a)}\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, e_{\lambda}\,{\lt}\,\tau_{0}^{-}\right)+\int_{0}^{a}\varphi(y;\;\theta,a)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, e_{\lambda}\,{\lt}\,\tau_{0}^{-}\right)\nonumber\\&\,=\,&\lambda\int_{a}^{\infty}\mathrm{e}^{-\theta (y-a)}\left(\mathrm{e}^{-\Phi_{\lambda}y}W_{\lambda}(x)-W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&&+\frac{\lambda^{2}}{\theta+\Phi_{\lambda}}\int_{0}^{a}\frac{W(y)}{Z(a,\Phi_{\lambda})}\left(\mathrm{e}^{-\Phi_{\lambda}y}W_{\lambda}(x)-W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&\,=\,&\lambda \,\mathrm{e}^{\theta a} W_{\lambda}(x)\int_{a}^{\infty}\mathrm{e}^{-(\theta +\Phi_{\lambda})y}\,\mathrm{d}y-\lambda\int_{a}^{\infty}\mathrm{e}^{-\theta (y-a)}W_{\lambda}(x-y)\mathrm{d}y\nonumber\\&&+\frac{\lambda^{2}}{\theta+\Phi_{\lambda}}\frac{W_{\lambda}(x)}{Z(a,\Phi_{\lambda})}\int_{0}^{a}W(y)\mathrm{e}^{-\Phi_{\lambda}y}\,\mathrm{d}y-\frac{\lambda^{2}}{\theta+\Phi_{\lambda}}\frac{1}{Z(a,\Phi_{\lambda})}\int_{0}^{a}W(y)W_{\lambda}(x-y),\mathrm{d}y\nonumber\\&\,=\,&\frac{\lambda \,\mathrm{e}^{-\Phi_{\lambda}a}W_{\lambda}(x)}{\theta +\Phi_{\lambda}}-\lambda\,\mathrm{e}^{-\theta (x-a)}\int_{0}^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\nonumber\\&&+\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{W_{\lambda}(x)\left(1-\mathrm{e}^{-\Phi_{\lambda}a}Z(a, \Phi_{\lambda})\right)}{Z(a,\Phi_{\lambda})}+\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{W_{a}^{(0,\lambda)}(x)-W_{\lambda}(x)}{Z(a,\Phi_{\lambda})}\nonumber\\&\,=\,&\frac{\lambda \,\mathrm{e}^{-\Phi_{\lambda}a}W_{\lambda}(x)}{\theta +\Phi_{\lambda}}-\lambda\,\mathrm{e}^{-\theta (x-a)}\int_{0}^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\nonumber\\&&+\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{W_{a}^{(0,\lambda)}(x)-\mathrm{e}^{-\Phi_{\lambda}a}W_{\lambda}(x)Z(a, \Phi_{\lambda})}{Z(a,\Phi_{\lambda})}\nonumber\\&\,=\,&\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{W_{a}^{(0,\lambda)}(x)}{Z(a,\Phi_{\lambda})}-\lambda\,\mathrm{e}^{-\theta (x-a)}\int_{0}^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y,\quad x\,{\gt}\,a,\nonumber\end{eqnarray}

where we observe that the integral representation on the right-hand side of the first equality arises from conditioning on the first Poisson arrival time, and (24) employed in the derivation may be regarded as a special case of (6). This completes the proof.

Proof. It can be found in Theorem 3.2 of [Reference Albrecher, Ivanovs and Zhou5] that

\begin{eqnarray}\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{0}^{-}\big)&\,=\,&\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{Z(x,\Phi_{\lambda})}{\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})},\quad x\leq a.\nonumber\end{eqnarray}

For $x\,{\gt}\, a$ , we have

\begin{eqnarray}&&\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{0}^{-}\big)\nonumber\\&\,=\,&\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; \tau_{0}^{-}\,{\lt}\,T_{a}^{+}\,{\lt}\,T_{0}^{-}\big)+\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{0}^{-}\big)\nonumber\\&\,=\,&\mathrm{E}_{x}\big(\mathrm{E}_{X(\tau_{0}^{-})}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;T_{a}^{+}\,{\lt}\,T_{0}^{-}\big);\;\tau_{0}^{-}\,{\lt}\,T_{a}^{+}\big)+\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{0}^{-}\big)\nonumber\\&\,=\,&\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{1}{\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})}\mathrm{E}_{x}\big(\mathrm{e}^{\Phi_{\lambda}X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,T_{a}^{+}\big)+\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{0}^{-}\big).\nonumber\end{eqnarray}

By Lemmas 5.1 and 5.2 we have

\begin{eqnarray}\mathrm{E}_{x}\big(\mathrm{e}^{\Phi_{\lambda}X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,T_{a}^{+}\big)&\,=\,&Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})+\lim\limits_{\theta\rightarrow\Phi_{\lambda}}\frac{W_{a}^{(0,\lambda)}(x)}{\theta-\Phi_{\lambda}}\left(\lambda\frac{Z(a,\theta)}{Z(a,\Phi_{\lambda})}-\psi(\theta)\right)\nonumber\\&\,=\,&Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})+\frac{W_{a}^{(0,\lambda)}(x)}{Z(a,\Phi_{\lambda})}\lim\limits_{\theta\rightarrow\Phi_{\lambda}}\frac{\psi(\Phi_{\lambda})Z(a,\theta)-\psi(\theta)Z(a,\Phi_{\lambda})}{\theta-\Phi_{\lambda}}\nonumber\\&\,=\,&Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})-\frac{W_{a}^{(0,\lambda)}(x)}{Z(a,\Phi_{\lambda})}\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda}),\nonumber\end{eqnarray}

and

\begin{eqnarray}\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{0}^{-}\big)&\,=\,&-\lambda\,\mathrm{e}^{-\theta (x-a)}\int_{0}^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y+\frac{\lambda W_{a}^{(0,\lambda)}(x)}{\left(\theta+\Phi_{\lambda}\right)Z(a,\Phi_{\lambda})}.\nonumber\end{eqnarray}

Therefore, for $x\,{\gt}\, a$ we have

\begin{eqnarray}\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{0}^{-}\big)&\,=\,&\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})}{\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})}-\lambda\,\mathrm{e}^{-\theta (x-a)}\int_{0}^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y.\nonumber\end{eqnarray}

It remains to prove (27). By (1) and (3) we have

(28) \begin{eqnarray}&&W_{a+u}^{(0,\lambda)}(x+u)\nonumber\\&\,=\,&W_{\lambda}(x+u)-\lambda\int_0^{a+u}W_{\lambda}(x+u-y)W(y)\mathrm{d}y\nonumber\\&\,=\,&W_{\lambda}(x+u)+\lambda\left(\int_0^{x-a}W_{\lambda}(y)W(x+u-y)\mathrm{d}y-\int_0^{x+u}W_{\lambda}(y)W(x+u-y)\mathrm{d}y\right)\nonumber\\&\,=\,&W_{\lambda}(x+u)-\left(W_{\lambda}(x+u)-W(x+u)\right)+\lambda\int_0^{x-a}W(x+u-y)W_{\lambda}(y)\mathrm{d}y\nonumber\\&\,=\,&W(x+u)+\lambda\int_0^{x-a}W(x+u-y)W_{\lambda}(y)\mathrm{d}y.\end{eqnarray}

It can be found in (6) of [Reference Albrecher, Ivanovs and Zhou5] and (3) of [Reference Czarna and Renaud10] that

\begin{eqnarray}&&\lim_{a\rightarrow\infty}\frac{Z(a,\theta)}{W(a)}=\frac{\psi(\theta)}{\theta-\Phi},\quad\lim_{x\rightarrow\infty}\frac{W_{\lambda}(x+y)}{W_{\lambda}(x)}=\mathrm{e}^{\Phi_{\lambda}y},\nonumber\end{eqnarray}

which imply

\begin{eqnarray}&&\lim_{u\rightarrow\infty}\frac{W(x+u)}{Z(a+u,\Phi_{\lambda})}=\frac{\Phi_{\lambda}-\Phi}{\lambda}\mathrm{e}^{\Phi(x-a)},\quad\lim_{u\rightarrow\infty}\frac{W(x+u-y)}{Z(a+u,\Phi_{\lambda})}=\frac{\Phi_{\lambda}-\Phi}{\lambda}\mathrm{e}^{\Phi(x-a-y)},\nonumber\end{eqnarray}

then it follows from (28) that

\begin{eqnarray}\lim_{u\rightarrow\infty}\frac{W_{a+u}^{(0,\lambda)}(x+u)}{Z(a+u,\Phi_{\lambda})}&\,=\,&\lim_{u\rightarrow\infty}\left(\frac{W(x+u)}{Z(a+u,\Phi_{\lambda})}+\lambda\int_0^{x-a}\frac{W(x+u-y)}{Z(a+u,\Phi_{\lambda})}W_{\lambda}(y)\mathrm{d}y\right)\nonumber\\&\,=\,&\frac{\Phi_{\lambda}-\Phi}{\lambda}\mathrm{e}^{\Phi(x-a)}+\lambda\int_0^{x-a}\frac{\Phi_{\lambda}-\Phi}{\lambda}\mathrm{e}^{\Phi(x-a-y)}W_{\lambda}(y)\mathrm{d}y,\nonumber\end{eqnarray}

which together with Lemma 5.2 further leads to

\begin{eqnarray}\mathrm{E}_{x}\big(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\infty\big)&\,=\,&\lim_{u\rightarrow\infty}\varphi(x+u;\;\theta,a+u)\nonumber\\&\,=\,&-\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y+\frac{\Phi_{\lambda}-\Phi}{\theta+\Phi_{\lambda}}Z_{\lambda}(x-a,\Phi).\nonumber\end{eqnarray}

This completes the proof.

Proof. From Theorem 3.2 in [Reference Albrecher, Ivanovs and Zhou5] it follows that

\begin{eqnarray}\mathrm{E}_{x}\big(\mathrm{e}^{\theta X(T_{0}^{-})};\; T_{0}^{-}\,{\lt}\,T_{a}^{+}\big)&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left(Z(x,\theta)-Z(x,\Phi_{\lambda})\frac{\widetilde{Z}(a,\Phi_{\lambda}, \theta)}{\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})}\right)\!,\quad x\leq a.\nonumber\end{eqnarray}

By conditioning on the first Poisson observation time and using (6), for $x\,{\gt}\,a$ , we have

\begin{eqnarray}&&\mathrm{E}_{x}\big(\mathrm{e}^{\theta X(T_{0}^{-})};\;T_{0}^{-}\,{\lt}\,T_{a}^{+}\big)\nonumber\\&\,=\,&\mathrm{E}_{x}\big(\mathrm{e}^{\theta X(T_{0}^{-})};\;T_{0}^{-}=e_{\lambda}\,{\lt}\, T_{a}^{+}\big)+\mathrm{E}_{x}\big(\mathrm{e}^{\theta X(T_{0}^{-})};\;e_{\lambda}\,{\lt}\,T_{0}^{-}\,{\lt}\,T_{a}^{+}\big)\nonumber\\&\,=\,&\int_{-\infty}^{0}\mathrm{e}^{\theta y}\mathrm{P}_{x}\left(X(e_{\lambda})\in \mathrm{d}y\right)+\int_{0}^{a}\mathrm{E}_{y}\big(\mathrm{e}^{\theta X(T_{0}^{-})};\;T_{0}^{-}\,{\lt}\,T_{a}^{+}\big)\mathrm{P}_{x}\left(X(e_{\lambda})\in \mathrm{d}y\right)\nonumber\\&\,=\,&\lambda\int_{-\infty}^{0}\mathrm{e}^{\theta y}\bigg(\frac{\mathrm{e}^{\Phi_{\lambda}(x-y)}}{\psi'(\Phi_{\lambda})}-W_{\lambda}(x-y)\bigg)\mathrm{d}y\nonumber\\&&+\frac{\lambda^{2}}{\lambda-\psi(\theta)}\int_{0}^{a}\left(Z(y,\theta)-Z(y,\Phi_{\lambda})\frac{\widetilde{Z}(a,\Phi_{\lambda}, \theta)}{\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})}\right)\left(\frac{\mathrm{e}^{\Phi_{\lambda}(x-y)}}{\psi'(\Phi_{\lambda})}-W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&\,=\,&\frac{\lambda}{\psi'(\Phi_{\lambda})}\frac{\mathrm{e}^{\Phi_{\lambda}x}}{\theta-\Phi_{\lambda}}-\lambda\frac{Z_{\lambda}(x,\theta)}{\psi(\theta)-\lambda}\nonumber\\&&+\frac{\lambda^{2}}{\lambda-\psi(\theta)}\frac{\mathrm{e}^{\Phi_{\lambda}x}}{\psi'(\Phi_{\lambda})}\frac{\mathrm{e}^{-\Phi_{\lambda}a}Z(a,\theta)-1}{\theta-\Phi_{\lambda}}+\frac{\lambda}{\lambda-\psi(\theta)}\frac{\mathrm{e}^{\Phi_{\lambda}x}}{\psi'(\Phi_{\lambda})}\frac{\psi(\theta)\left(1-\mathrm{e}^{-\Phi_{\lambda}a}Z(a,\Phi_{\lambda})\right)}{\theta-\Phi_{\lambda}}\nonumber\\&&+\frac{\lambda}{\lambda-\psi(\theta)}\left(Z_{a}^{(0,\lambda)}(x,\theta)-Z_{\lambda}(x,\theta)\right)-\frac{\lambda}{\lambda-\psi(\theta)}\frac{\widetilde{Z}(a,\Phi_{\lambda}, \theta)}{\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})}\big(Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})-\mathrm{e}^{\Phi_{\lambda}x}\big)\nonumber\\&&-\frac{\lambda}{\lambda-\psi(\theta)}\frac{\widetilde{Z}(a,\Phi_{\lambda}, \theta)}{\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})}\frac{\mathrm{e}^{\Phi_{\lambda}x}}{\psi'(\Phi_{\lambda})}\left({-}\mathrm{e}^{-\Phi_{\lambda}a}\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})+\psi'(\Phi_{\lambda})\right)\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left(Z_{a}^{(0,\lambda)}(x,\theta)-\frac{\widetilde{Z}(a,\Phi_{\lambda}, \theta)}{\widetilde{Z}(a,\Phi_{\lambda}, \Phi_{\lambda})}Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})\right)\!.\nonumber\end{eqnarray}

This completes the proof.

Proof. For $x\leq 0,$ we have

\begin{eqnarray}\rho(x;\;\lambda, a, b)=\mathrm{E}_{x}\bigg(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t}\bigg)=\mathrm{E}_{x}\Big(\mathrm{e}^{-\lambda\tau_{0}^{+}}\Big)\rho(0;\;\lambda, a, b)=\mathrm{e}^{\Phi_{\lambda}x}\rho(0;\;\lambda, a, b).\nonumber\end{eqnarray}

For $x\in[0,a),$

\begin{eqnarray}&&\rho(x;\;\lambda, a, b)=\mathrm{E}_{x}\bigg(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t}\bigg)\nonumber\\&\,=\,&\mathrm{E}_{x}\bigg(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t};\;\tau_{b}^{+}\,{\lt}\,\tau_{0}^{-}\bigg)+\mathrm{E}_{x}\bigg(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t};\;\tau_{0}^{-}\,{\lt}\,\tau_{b}^{+}\bigg)\nonumber\\&\,=\,&\mathrm{P}_{x}\big(\tau_{a}^{+}\,{\lt}\,\tau_{0}^{-}\big)\mathrm{E}_{a}\bigg(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t};\;\tau_{b}^{+}\,{\lt}\,\tau_{0}^{-}\bigg)\nonumber \end{eqnarray}

\begin{eqnarray}&&+\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t};\;\tau_{0}^{-}\,{\lt}\,\tau_{a}^{+}\right)+\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t};\;\tau_{a}^{+}\,{\lt}\,\tau_{0}^{-}\,{\lt}\,\tau_{b}^{+}\right)\nonumber\\&\,=\,&\mathrm{P}_{x}\left(\tau_{a}^{+}\,{\lt}\,\tau_{0}^{-}\right)\mathrm{E}_{a}\left(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t};\;\tau_{b}^{+}\,{\lt}\,\tau_{0}^{-}\right)\nonumber\\&&+\mathrm{E}_{x}\left(\rho(X(\tau_{0}^{-});\;\lambda, a, b);\;\tau_{0}^{-}\,{\lt}\,\tau_{a}^{+}\right)+\mathrm{P}_{x}\left(\tau_{a}^{+}\,{\lt}\,\tau_{0}^{-}\right)\mathrm{E}_{a}\left(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t};\;\tau_{0}^{-}\,{\lt}\,\tau_{b}^{+}\right)\nonumber\\&\,=\,&\mathrm{P}_{x}\left(\tau_{a}^{+}\,{\lt}\,\tau_{0}^{-}\right)\mathrm{E}_{a}\left(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t}\right)\nonumber\\&&+\mathrm{E}_{x}\left(\mathrm{e}^{\Phi_{\lambda}X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,\tau_{a}^{+}\right)\rho(0;\;\lambda, a, b)\nonumber\\&\,=\,&\frac{W(x)}{W(a)}\rho(a;\;\lambda, a, b)+\left(Z(x,\Phi_{\lambda})-\frac{Z(a,\Phi_{\lambda})}{W(a)}W(x)\right)\rho(0;\;\lambda, a, b).\nonumber \end{eqnarray}

Recalling that $Z(x,\Phi_{\lambda})=\mathrm{e}^{\Phi_{\lambda}x}$ and $W(x)=0$ for $x\,{\lt}\,0$ , we have for $x\in [a,b]$ that

\begin{eqnarray}\rho(x;\;\lambda, a, b)&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{a}^{-}}\rho(X(\tau_{a}^{-});\;\lambda, a, b);\;\tau_{a}^{-}\,{\lt}\,\tau_{b}^{+}\right)+\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}\nonumber\\&\,=\,&\left(\frac{\rho(a;\;\lambda, a, b)}{W(a)}-\frac{Z(a,\Phi_{\lambda})}{W(a)}\rho(0;\;\lambda, a, b)\right)\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{a}^{-}}W(X(\tau_{a}^{-}));\;\tau_{a}^{-}\,{\lt}\,\tau_{b}^{+}\right)\nonumber\\&&+\rho(0;\;\lambda, a, b)\,\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{a}^{-}}Z(X(\tau_{a}^{-}),\Phi_{\lambda});\;\tau_{a}^{-}\,{\lt}\,\tau_{b}^{+}\right)+\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}.\nonumber\end{eqnarray}

The scale function $Z(x,\Phi_{\lambda})$ can be rewritten as

$$Z(x,\Phi_{\lambda})\;:\!=\;\mathrm{E}_{x}\left(\mathrm{e}^{\Phi_{\lambda} X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,\infty\right)+\frac{\lambda}{\Phi_{\lambda}-\Phi}W(x),$$

with $\mathrm{E}_{x}\left(\mathrm{e}^{\Phi_{\lambda} X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,\infty\right)$ being the same form as (14) in [Reference Loeffen, Renaud and Zhou22], and W(x) satisfying (13) in [Reference Loeffen, Renaud and Zhou22]. By Lemma 2.1 in [Reference Loeffen, Renaud and Zhou22], we know that

\begin{eqnarray}\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{a}^{-}}Z(X(\tau_{a}^{-}),\Phi_{\lambda});\;\tau_{a}^{-}\,{\lt}\,\tau_{b}^{+}\right)&\,=\,&Z(x,\Phi_{\lambda})+\lambda \int_{a}^{x}W_{\lambda}(x-y)Z(y,\Phi_{\lambda})\mathrm{d}y\nonumber\\&&-\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}\left(Z(b,\Phi_{\lambda})+\lambda \int_{a}^{b}W_{\lambda}(b-y)Z(y,\Phi_{\lambda})\mathrm{d}y\!\right)\nonumber\\&\,=\,&Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})-\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda}).\nonumber\end{eqnarray}

In addition, it follows from [Reference Li and Zhou19] that

\begin{eqnarray}\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{a}^{-}}W(X(\tau_{a}^{-}));\;\tau_{a}^{-}\,{\lt}\,\tau_{b}^{+}\right)=W_{a}^{(0,\lambda)}(x)-\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}W_{a}^{(0,\lambda)}(b).\nonumber\end{eqnarray}

Therefore, for $x\in [a,b]$ , we have

\begin{eqnarray}\rho(x;\;\lambda, a, b)&\,=\,&\left(\frac{\rho(a;\;\lambda, a, b)}{W(a)}-\frac{Z(a,\Phi_{\lambda})}{W(a)}\rho(0;\;\lambda, a, b)\right)\left(W_{a}^{(0,\lambda)}(x)-\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}W_{a}^{(0,\lambda)}(b)\right)\nonumber\\&&+\rho(0;\;\lambda, a, b)\,\left(Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})-\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda})\right)+\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}.\nonumber\end{eqnarray}

It remains to solve $\rho(0;\;\lambda, a, b)$ and $\rho(a;\;\lambda, a, b)$ . For this purpose, we write $0\,{\lt}\,e_{\lambda}(1)\,{\lt}$ $e_{\lambda}(2)\,{\lt}\,\cdots$ and $0\,{\lt}\,\overline{e}_{\lambda}(1)\,{\lt}\,\overline{e}_{\lambda}(2)\,{\lt}\,\cdots$ for all the arrival times of two independent Poisson processes with the same rate $\lambda$ . We also assume that both Poisson processes are independent of process X. Then for two independent exponential random variables $e_{\lambda}$ and $\overline{e}_{\lambda}$ with same rate $\lambda$ , we have

(29) \begin{eqnarray}&&\rho(0;\;\lambda, a, b)=\mathrm{E}\left(\mathrm{e}^{-\lambda\tau_{b}^{+}+\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{[0,a]}(X(t))\mathrm{d}t}\right)\nonumber\\[5pt]&\,=\,&\mathrm{E}\left(\mathrm{e}^{-\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{({-}\infty,0)}(X(t))\mathrm{d}t-\lambda\int_{0}^{\tau_{b}^{+}}\textbf{1}_{(a,b]}(X(t))\mathrm{d}t}\right)\nonumber\\[5pt]&\,=\,&\mathrm{P}\left(\{e_{\lambda}(i)\}\cap\{s\;:\;s\,{\lt}\,\tau_{b}^{+},X(s)\in({-}\infty,0)\}=\{\overline{e}_{\lambda}(i)\}\cap\{s\;:\;s\,{\lt}\,\tau_{b}^{+},X(s)\in(a,b)\}=\emptyset\right)\nonumber\\[5pt]&\,=\,&\mathrm{P}\left(\tau_{b}^{+}\,{\lt}\,e_{\lambda}\wedge \overline{e}_{\lambda}\right)+\int_{0}^{b}\mathrm{P}\left(e_{\lambda}\,{\lt}\,\tau_{b}^{+}\wedge \overline{e}_{\lambda},X(e_{\lambda})\in\mathrm{d} y\right)\rho(y;\;\lambda, a, b)\nonumber\\[5pt]&&+\int_{-\infty}^{a}\mathrm{P}\left(\overline{e}_{\lambda}\,{\lt}\,\tau_{b}^{+}\wedge e_{\lambda},X(\overline{e}_{\lambda})\in\mathrm{d} y\right)\rho(y;\;\lambda, a, b)\nonumber\\[5pt]&\,=\,&\mathrm{e}^{-\Phi_{2\lambda}b}+\lambda\int_{0}^{b}\int_{0}^{\infty}\mathrm{e}^{-2\lambda t}\mathrm{P}\left(t\,{\lt}\,\tau_{b}^{+},X(t)\in\mathrm{d} y\right)\rho(y;\;\lambda, a, b)\mathrm{d}t\nonumber\\[5pt]&&+\lambda\int_{-\infty}^{a}\int_{0}^{\infty}\mathrm{e}^{-2\lambda t}\mathrm{P}\left(t\,{\lt}\,\tau_{b}^{+},X(t)\in\mathrm{d} y\right)\rho(y;\;\lambda, a, b)\mathrm{d}t\nonumber\\[5pt]&\,=\,&\mathrm{e}^{-\Phi_{2\lambda}b}+\frac{1}{2}\int_{0}^{b}\mathrm{P}\left(e_{2\lambda}\,{\lt}\,\tau_{b}^{+},X(e_{2\lambda})\in\mathrm{d} y\right)\rho(y;\;\lambda, a, b)\nonumber\\[5pt]&&+\frac{1}{2}\int_{-\infty}^{a}\mathrm{P}\left(\overline{e}_{2\lambda}\,{\lt}\,\tau_{b}^{+},X(\overline{e}_{2\lambda})\in\mathrm{d} y\right)\rho(y;\;\lambda, a, b)\nonumber\\[5pt]&\,=\,&\mathrm{e}^{-\Phi_{2\lambda}b}+\lambda\mathrm{e}^{-\Phi_{2\lambda}b}\int_{a}^{b}W_{2\lambda}(b-x)\rho(x;\;\lambda, a, b)\mathrm{d} x\nonumber\\[5pt]&&+2\lambda\mathrm{e}^{-\Phi_{2\lambda}b}\int_{0}^{a} W_{2\lambda}(b-x)\rho(x;\;\lambda, a, b)\mathrm{d} x\nonumber\\[5pt]&&+\lambda\int_{-\infty}^{0}\left(\mathrm{e}^{-\Phi_{2\lambda}b}W_{2\lambda}(b-x)-W_{2\lambda}({-}x)\right)\rho(x;\;\lambda, a, b)\mathrm{d} x\nonumber\\[5pt]&\,=\,&\mathrm{e}^{-\Phi_{2\lambda}b}\frac{W_{2\lambda}(b-a)}{W_{\lambda}(b-a)}+\mathrm{e}^{-\Phi_{2\lambda}b}\rho(a;\;\lambda, a, b)\left(\frac{W_{2\lambda}(b)}{W(a)}-\frac{W_{a}^{(0,\lambda)}(b)}{W(a)}\frac{W_{2\lambda}(b-a)}{W_{\lambda}(b-a)}\right)\nonumber\\[5pt]&&+\mathrm{e}^{-\Phi_{2\lambda}b}\rho(0;\;\lambda, a, b)\left(\mathrm{e}^{\Phi_{2\lambda}b}-Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda})\frac{W_{2\lambda}(b-a)}{W_{\lambda}(b-a)}\right)\nonumber\\[5pt]&&-\mathrm{e}^{-\Phi_{2\lambda}b}\rho(0;\;\lambda, a, b)\ Z(a,\Phi_{\lambda})\left(\frac{W_{2\lambda}(b)}{W(a)}-\frac{W_{a}^{(0,\lambda)}(b)}{W(a)}\frac{W_{2\lambda}(b-a)}{W_{\lambda}(b-a)}\right)\!,\end{eqnarray}

and

(30) \begin{eqnarray}&&\rho(a;\;\lambda, a, b)\nonumber\\&\,=\,&\mathrm{e}^{-\Phi_{2\lambda}b}+\lambda\mathrm{e}^{-\Phi_{2\lambda}b}\int_{a}^{b}W_{2\lambda}(b-x)\rho(x;\;\lambda, a, b)\mathrm{d} x\nonumber\\&&+2\lambda\int_{0}^{a}\left(\mathrm{e}^{-\Phi_{2\lambda}b}W_{2\lambda}(b-x)-W_{2\lambda}(a-x)\right)\rho(x;\;\lambda, a, b)\mathrm{d} x\nonumber\\&&+\lambda\int_{-\infty}^{0}\left(\mathrm{e}^{-\Phi_{2\lambda}b}W_{2\lambda}(b-x)-W_{2\lambda}(a-x)\right)\rho(x;\;\lambda, a, b)\mathrm{d} x\nonumber\\&\,=\,&\mathrm{e}^{-\Phi_{2\lambda}b}\frac{W_{2\lambda}(b-a)}{W_{\lambda}(b-a)}\nonumber\\&&+\mathrm{e}^{-\Phi_{2\lambda}b}\rho(a;\;\lambda, a, b)\left(\frac{W_{2\lambda}(b)}{W(a)}-\frac{W_{a}^{(0,\lambda)}(b)}{W(a)}\frac{W_{2\lambda}(b-a)}{W_{\lambda}(b-a)}-\mathrm{e}^{\Phi_{2\lambda}b}\frac{W_{2\lambda}(a)}{W(a)}+\mathrm{e}^{\Phi_{2\lambda}b}\right)\nonumber\\&&+\mathrm{e}^{-\Phi_{2\lambda}b}\rho(0;\;\lambda, a, b)\left({-}Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda})\frac{W_{2\lambda}(b-a)}{W_{\lambda}(b-a)}+\mathrm{e}^{\Phi_{2\lambda}b}\ Z(a,\Phi_{\lambda})\frac{W_{2\lambda}(a)}{W(a)}\right)\nonumber\\&&-\mathrm{e}^{-\Phi_{2\lambda}b}\rho(0;\;\lambda, a, b)\ Z(a,\Phi_{\lambda})\left(\frac{W_{2\lambda}(b)}{W(a)}-\frac{W_{a}^{(0,\lambda)}(b)}{W(a)}\frac{W_{2\lambda}(b-a)}{W_{\lambda}(b-a)}\right)\!.\end{eqnarray}

Solving (29) and (30), we get

\begin{eqnarray}\rho(a;\;\lambda, a, b)&\,=\,&\ Z(a,\Phi_{\lambda})\rho(0;\;\lambda, a, b),\nonumber\\\rho(0;\;\lambda, a, b)&\,=\,&\frac{1}{Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda})},\nonumber\\\rho(a;\;\lambda, a, b)&\,=\,&\frac{Z(a,\Phi_{\lambda})}{Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda})}.\nonumber\end{eqnarray}

Therefore, for $x\in ({-}\infty,a),$ we have

\begin{eqnarray}\rho(x;\;\lambda, a, b)&\,=\,&\frac{W(x)}{W(a)}\rho(a;\;\lambda, a, b)+\left(\!Z(x,\Phi_{\lambda})-\frac{Z(a,\Phi_{\lambda})}{W(a)}W(x)\!\right)\rho(0;\;\lambda, a, b)=\frac{Z(x,\Phi_{\lambda})}{Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda})},\nonumber\end{eqnarray}

and, for $x\in [a,b],$ we have

\begin{eqnarray}\rho(x;\;\lambda, a, b)&\,=\,&\left(\frac{\rho(a;\;\lambda, a, b)}{W(a)}-\frac{Z(a,\Phi_{\lambda})}{W(a)}\rho(0;\;\lambda, a, b)\right)\left(W_{a}^{(0,\lambda)}(x)-\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}W_{a}^{(0,\lambda)}(b)\right)\nonumber\\&&+\rho(0;\;\lambda, a, b)\,\left(Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})-\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda})\right)+\frac{W_{\lambda}(x-a)}{W_{\lambda}(b-a)}\nonumber\\&\,=\,&\frac{Z_{a}^{(0,\lambda)}(x,\Phi_{\lambda})}{Z_{a}^{(0,\lambda)}(b,\Phi_{\lambda})}.\nonumber\end{eqnarray}

This completes the proof.

Proof. We first prove that, under the condition $\lim_{z\rightarrow\infty}\overline{\xi}(z)=\infty$ , we have

(31) \begin{eqnarray}\lim\limits_{x\rightarrow\infty}f_{1}(x;\; \theta, a)&\,=\,&0,\end{eqnarray}

(32) \begin{eqnarray}\lim\limits_{x\rightarrow\infty}f_{2}(x;\; \theta, a)&\,=\,&\textbf{1}_{\{\theta=0\}},\end{eqnarray}

(33) \begin{eqnarray}\lim\limits_{x\rightarrow\infty}f_{3}(x;\; \theta, a) &\,=\,& 0.\end{eqnarray}

In fact, stopping times $\tau_{\xi}$ and $T_{a}^{+}$ can be rewritten as

\begin{align} \tau_{\xi}=& \inf\{t\geq0;\; \overline{Y}(t)-Y(t)\,{\gt}\,\overline{\xi}(x+\overline{Y}(t))\}=\inf\big\{t\geq0;\, Y(t)\,{\lt}\,\zeta_{x}(\overline{Y}(t))\big\},\nonumber \\ T_{\xi}=& \inf\big\{T_{i}; \,\overline{Y}(T_{i})-Y(T_{i})\,{\gt}\,\overline{\xi}(x+\overline{Y}(T_{i}))\big\}=\inf\big\{T_{i};\, Y(T_{i})\,{\lt}\,\zeta_{x}(\overline{Y}(T_{i}))\big\},\nonumber \\ T_{a}^{+}=&\inf\big\{T_{i};\, Y(T_{i})\,{\gt}\,a-x\big\},\nonumber\end{align}

with $Y(t)\;:\!=\;X(t)-x$ , $\overline{Y}(t)\;:\!=\;\sup_{s\leq t}Y(s)$ , and $\zeta_{x}(y)\;:\!=\;\xi(x+y)-x$ . It is not hard to see that $y\mapsto \zeta_{x}(y)$ is a drawdown function with $\lim_{x\rightarrow\infty}\zeta_{x}(y)=\lim_{x\rightarrow\infty}\left(y-\overline{\xi}(x+y)\right)=-\infty$ uniformly for all $y\in\mathbb{R}_{+}$ , and

$$\mathrm{P}_{0}\left(\lim\limits_{x\rightarrow\infty}\tau_{\zeta_{x}}=\infty,\,\lim\limits_{x\rightarrow\infty}T_{\zeta_{x}}=\infty,\,\lim\limits_{x\rightarrow\infty}T_{a-x}^{+}=T_{1},\, \lim\limits_{x\rightarrow\infty}\tau_{a-x}^{+}=0\right)=1.$$

Hence,

\begin{align}\lim\limits_{x\rightarrow\infty}\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,\tau_{a}^{+}\right)\leq &\lim\limits_{x\rightarrow\infty}\mathrm{P}_{0}\left(T_{\zeta_{x}}\,{\lt}\,\tau_{a-x}^{+}\right)=0,\nonumber\\\lim\limits_{x\rightarrow\infty}\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,T_{a}^{+}\right)\leq &\lim\limits_{x\rightarrow\infty}\mathrm{P}_{0}\left(\tau_{\zeta_{x}}\,{\lt}\,T_{a-x}^{+}\right)=0,\nonumber\\\lim\limits_{x\rightarrow\infty}\mathrm{E}_{x}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{\xi}\right)=&\lim\limits_{x\rightarrow\infty}\mathrm{e}^{-\theta (x-a)}\mathrm{E}_{0}\left(\mathrm{e}^{-\theta X(T_{a-x}^{+})};\; T_{a-x}^{+}\,{\lt}\,\tau_{\zeta_{x}}\right)\nonumber\\=&\begin{cases} 1,&\theta=0,\\ \lim\limits_{x\rightarrow\infty}\mathrm{e}^{-\theta (x-a)}\mathrm{E}_{0}\left(\mathrm{e}^{-\theta X(T_{1})}\right)=0,&\theta\,{\gt}\,0,\end{cases}\nonumber\end{align}

these prove (31), (32), and (33). Now, define $\kappa_{b}^{+}$ to be the first up-crossing time of level b after the drawdown time $\tau_{\xi}$ as follows:

$$\kappa_{b}^{+}\;:\!=\;\inf\{t\geq\tau_{\xi};\; X(t)\geq b\}.$$

Then we have

\begin{eqnarray*}&&f_{1}(x;\;\theta,a)\nonumber\\&\;:\!=\;&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,\tau_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; \tau_{\xi}\leq T_{\xi}\,{\lt}\,\kappa_{\overline{X}(\tau_{\xi})}^{+}\,{\lt}\,\tau_{a}^{+}\right)\nonumber\\&&+\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; \tau_{\xi}\leq\kappa_{\overline{X}(\tau_{\xi})}^{+} \,{\lt}\,T_{\xi}\,{\lt}\,\tau_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left( \mathrm{E}_{x}\left(\left.\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\;T_{\xi}\,{\lt}\,\kappa_{\overline{X}(\tau_{\xi})}^{+}\right|\mathcal{F_{\tau_{\xi}}}\right);\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&&+\mathrm{E}_{x}\left(\mathrm{E}_{x}\left(\left.\mathrm{E}_{x}\left(\left.\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,\tau_{a}^{+}\right|\mathcal{F}_{\kappa_{\overline{X}(\tau_{\xi})}^{+}}\right);\; \kappa_{\overline{X}(\tau_{\xi})}^{+}\,{\lt}\,T_{\xi}\right|\mathcal{F_{\tau_{\xi}}}\right);\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\left.\left[\mathrm{E}_{X(\tau_{\xi})}\left(\mathrm{e}^{\theta \left(X(T_{\xi(b)}^{-})-\xi(b)\right)};\; T_{\xi(b)}^{-}\,{\lt}\,\tau_{b}^{+}\right)\right]\right|_{b=\overline{X}(\tau_{\xi})};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&&+\mathrm{E}_{x}\left(\left.\mathrm{E}_{x}\left(\mathrm{E}_{\overline{X}(\tau_{\xi})}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi})\right))};\; T_{\xi}\,{\lt}\,\tau_{a}^{+}\right);\; \kappa_{\overline{X}(\tau_{\xi})}^{+}\,{\lt}\,T_{\xi}\right|\mathcal{F_{\tau_{\xi}}}\right);\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\end{eqnarray*}

(34) \begin{eqnarray}&\,=\,&\mathrm{E}_{x}\left(\left.\left[\mathrm{E}_{X(\tau_{\xi})}\left(\mathrm{e}^{\theta \left(X(T_{\xi(b)}^{-})-\xi(b)\right)};\; T_{\xi(b)}^{-}\,{\lt}\,\tau_{b}^{+}\right)\right]\right|_{b=\overline{X}(\tau_{\xi})};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&&+\mathrm{E}_{x}\left(f_{1}(\overline{X}(\tau_{\xi});\;\theta,a)\left.\left[\mathrm{P}_{X(\tau_{\xi})}\left( \tau_{b}^{+}\,{\lt}\,T_{\xi(b)}^{-}\right)\right]\right|_{b=\overline{X}(\tau_{\xi})};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\!.\end{eqnarray}

As $X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\leq 0$ , then by (3) and (15) in [Reference Albrecher, Ivanovs and Zhou5], we get

(35) \begin{eqnarray}&&\left.\mathrm{E}_{X(\tau_{\xi})}\left(\mathrm{e}^{\theta \left(X(T_{\xi(b)}^{-})-\xi(b)\right)};\; T_{\xi(b)}^{-}\,{\lt}\,\tau_{b}^{+}\right)\right|_{b=\overline{X}(\tau_{\xi})}\nonumber\\&\,=\,&\left.\mathrm{E}_{X(\tau_{\xi})-\xi(b)}\left(\mathrm{e}^{\theta X(T_{0}^{-})};\; T_{0}^{-}\,{\lt}\,\tau_{b-\xi(b)}^{+}\right)\right|_{b=\overline{X}(\tau_{\xi})}\nonumber\\&\,=\,&\frac{\lambda }{\lambda-\psi(\theta)}\left(\mathrm{e}^{\theta X(\tau_{\xi})}\mathrm{e}^{-\theta\xi(\overline{X}(\tau_{\xi}))}-\mathrm{e}^{\Phi_{\lambda}X(\tau_{\xi})}\mathrm{e}^{-\Phi_{\lambda}\xi(\overline{X}(\tau_{\xi}))}\frac{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\theta)}{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\Phi_{\lambda})}\right)\end{eqnarray}

and

(36) \begin{eqnarray}\left.\mathrm{P}_{X(\tau_{\xi})}\left( \tau_{b}^{+}\,{\lt}\,T_{\xi(b)}^{-}\right)\right|_{b=\overline{X}(\tau_{\xi})}&\,=\,&1-\left.\mathrm{P}_{X(\tau_{\xi})-\xi(b)}\left(T_{0}^{-}\,{\lt}\,\tau_{b-\xi(b)}^{+}\right)\right|_{b=\overline{X}(\tau_{\xi})}\nonumber\\&\,=\,&1-\frac{\lambda}{\lambda-\psi(0)}\left(1-\mathrm{e}^{\Phi_{\lambda}X(\tau_{\xi})}\mathrm{e}^{-\Phi_{\lambda}\xi(\overline{X}(\tau_{\xi}))}\frac{Z(\overline{\xi}(\overline{X}(\tau_{\xi})))}{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\Phi_{\lambda})}\right)\nonumber\\&\,=\,&\mathrm{e}^{\Phi_{\lambda}X(\tau_{\xi})}\mathrm{e}^{-\Phi_{\lambda}\xi(\overline{X}(\tau_{\xi}))}\frac{Z(\overline{\xi}(\overline{X}(\tau_{\xi})))}{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\Phi_{\lambda})}.\end{eqnarray}

Combining (34), (35), and (36), we have

(37) \begin{eqnarray}f_{1}(x;\;\theta,a)&\;:\!=\;&\frac{\lambda}{\lambda-\psi(\theta)}\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&&-\frac{\lambda}{\lambda-\psi(\theta)}\mathrm{E}_{x}\left(\mathrm{e}^{\Phi_{\lambda}\left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)}\frac{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\theta)}{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\Phi_{\lambda})};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&&+\mathrm{E}_{x}\left(\mathrm{e}^{\Phi_{\lambda}X(\tau_{\xi})}f_{1}(\overline{X}(\tau_{\xi});\;\theta,a)\mathrm{e}^{-\Phi_{\lambda}\xi(\overline{X}(\tau_{\xi}))}\frac{Z(\overline{\xi}(\overline{X}(\tau_{\xi})))}{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\Phi_{\lambda})};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\!.\end{eqnarray}

By the compensation formula in excursion theory (see Corollary 4.11 of [Reference Bertoin9], Theorem 4.4 of [Reference Kyprianou13], or Proposition 3.1 of [Reference Li, Vu and Zhou15]), (5), and (8), we have

\begin{eqnarray*}&&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\sum_{r\in[0,a-x]}\prod_{s\,{\lt}\, r}\textbf{1}_{\{\overline{\varepsilon}_{s}\leq \overline{\xi}(x+s)\}}\,\mathrm{e}^{\theta(\overline{\xi}(x+r)-\varepsilon_{r}(\chi^{+}_{\overline{\xi}(x+r)}))}\textbf{1}_{\{\overline{\varepsilon}_{r}\,{\gt}\, \overline{\xi}(x+r)\}}\right)\nonumber\\&\,=\,&\int_{0}^{a -x}\mathrm{P}_{x}(L_{r}^{-1}\,{\lt}\,\tau_{\xi})\, n\left(\mathrm{e}^{\theta(\overline{\xi}(x+r)-\varepsilon(\chi^{+}_{\overline{\xi}(x+r)}))} \textbf{1}_{\{\overline{\varepsilon}\,{\gt}\,\overline{\xi}(x+r)\}}\right)\mathrm{d}r\nonumber\\&\,=\,&\int_{0}^{a-x}\mathrm{e}^{-\int_{x}^{x+r}\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\mathrm{d}z} n\left(\mathrm{e}^{\theta(\overline{\xi}(x+r)-\varepsilon(\chi^{+}_{\overline{\xi}(x+r)}))} \textbf{1}_{\{\overline{\varepsilon}\,{\gt}\,\overline{\xi}(x+r)\}}\right)\mathrm{d}r\nonumber\end{eqnarray*}

(38) \begin{eqnarray}&\,=\,&\int_{x}^{a}\mathrm{e}^{-\int_{x}^{s}\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\mathrm{d}z} n\left(\mathrm{e}^{\theta(\overline{\xi}(s)-\varepsilon(\chi^{+}_{\overline{\xi}(s)}))};\; \overline{\varepsilon}\,{\gt}\,\overline{\xi}(s)\right)\mathrm{d}s\nonumber\\&\,=\,&\int_{x}^{a}\mathrm{e}^{-\int_{x}^{s}\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\,\mathrm{d}z}\left(\frac{W'(\overline{\xi}(s))}{W(\overline{\xi}(s))}Z(\overline{\xi}(s),\theta)-\theta Z(\overline{\xi}(s),\theta)+\psi(\theta)W(\overline{\xi}(s))\right)\mathrm{d}s,\end{eqnarray}

(39) \begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{\Phi_{\lambda}\left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)}\frac{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\theta)}{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\Phi_{\lambda})};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&\,=\,&\int_{x}^{a}\mathrm{e}^{-\int_{x}^{s}\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\mathrm{d}z}\frac{Z(\overline{\xi}(s),\theta)}{Z(\overline{\xi}(s),\Phi_{\lambda})}\nonumber\\&& \times\left(\frac{W'(\overline{\xi}(s))}{W(\overline{\xi}(s))}Z(\overline{\xi}(s),\Phi_{\lambda})-\Phi_{\lambda} Z(\overline{\xi}(s),\Phi_{\lambda})+\lambda W(\overline{\xi}(s))\right)\mathrm{d}s,\end{eqnarray}

and

(40) \begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{\Phi_{\lambda}X(\tau_{\xi})}f_{1}(\overline{X}(\tau_{\xi});\;\theta,a)\mathrm{e}^{-\Phi_{\lambda}\xi(\overline{X}(\tau_{\xi}))}\frac{Z(\overline{\xi}(\overline{X}(\tau_{\xi})))}{Z(\overline{\xi}(\overline{X}(\tau_{\xi})),\Phi_{\lambda})};\;\tau_{\xi}\,{\lt}\, \tau_{a}^{+}\right)\nonumber\\&\,=\,&\int_{x}^{a}\mathrm{e}^{-\int_{x}^{s}\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\,\mathrm{d}z}\mathrm{e}^{-\Phi_{\lambda}\xi(s)}\frac{Z(\overline{\xi}(s))}{Z(\overline{\xi}(s),\Phi_{\lambda})}f_{1}(s;\;\theta,a)\nonumber\\&& \times\mathrm{e}^{\Phi_{\lambda}\xi(s)}\left(\frac{W'(\overline{\xi}(s))}{W(\overline{\xi}(s))}Z(\overline{\xi}(s),\Phi_{\lambda})-\Phi_{\lambda} Z(\overline{\xi}(s),\Phi_{\lambda})+\lambda W(\overline{\xi}(s))\right)\mathrm{d}s.\end{eqnarray}

Plugging (38)–(40) into (37) and then differentiating both sides of the yielding equation with respect to x, we have

(41) \begin{eqnarray}&&f_{1}'(x;\;\theta,a)\nonumber\\&\;:\!=\;&f_{1}(x;\;\theta,a)\!\left(\frac{W'(\overline{\xi}(x))}{W(\overline{\xi}(x))}-\frac{Z(\overline{\xi}(x))}{Z(\overline{\xi}(x),\Phi_{\lambda})}\!\left(\frac{W'(\overline{\xi}(x))}{W(\overline{\xi}(x))}Z(\overline{\xi}(x),\Phi_{\lambda})-\Phi_{\lambda} Z(\overline{\xi}(x),\Phi_{\lambda})+\lambda W(\overline{\xi}(x))\!\right)\!\!\right)\nonumber\\&&-\frac{\lambda}{\lambda-\psi(\theta)}\left(\frac{W'(\overline{\xi}(x))}{W(\overline{\xi}(x))}Z(\overline{\xi}(x),\theta)-\theta Z(\overline{\xi}(x),\theta)+\psi(\theta)W(\overline{\xi}(x))\right)\nonumber\\&&+\frac{\lambda}{\lambda-\psi(\theta)}\frac{Z(\overline{\xi}(x),\theta)}{Z(\overline{\xi}(x),\Phi_{\lambda})}\left(\frac{W'(\overline{\xi}(x))}{W(\overline{\xi}(x))}Z(\overline{\xi}(x),\Phi_{\lambda})-\Phi_{\lambda} Z(\overline{\xi}(x),\Phi_{\lambda})+\lambda W(\overline{\xi}(x))\right)\nonumber\\&\,=\,&{f_{1}(x;\;\theta,a)\left(\Phi_{\lambda}-\frac{\lambda W(\overline{\xi}(x))}{Z(\overline{\xi}(x),\Phi_{\lambda})}\right)} {-\frac{\lambda}{\lambda-\psi(\theta)}\left({-}\theta Z(\overline{\xi}(x),\theta)+\psi(\theta)W(\overline{\xi}(x))\right)}\nonumber\\&&{+\frac{\lambda}{\lambda-\psi(\theta)}\frac{Z(\overline{\xi}(x),\theta)}{Z(\overline{\xi}(x),\Phi_{\lambda})}\left({-}\Phi_{\lambda} Z(\overline{\xi}(x),\Phi_{\lambda})+\lambda W(\overline{\xi}(x))\right)\!.}\end{eqnarray}

Solving (41) with boundary condition (31) yields (10).

Identity (9) for $\theta\,{\gt}\,\Phi$ follows immediately from (10) by letting a tend to $\infty$ ; by analytic continuation it is also true for any $\theta\geq0$ .

For $x\le a$ , by the Markov property of X and (5), we have

\begin{eqnarray}f_{2}(x;\;\theta,a)&\;:\!=\;&\mathrm{E}_{x}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{\xi}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{E}_{x}\left(\left.\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{\xi}\right|\mathcal{F}_{\tau_{a}^{+}}\right);\; \tau_{a}^{+}\,{\lt}\,\tau_{\xi}\right)\nonumber\\&\,=\,&\mathrm{P}_{x}\left(\tau_{a}^{+}\,{\lt}\,\tau_{\xi}\right)f_{2}(a;\;\theta,a)\nonumber\\&\,=\,&\mathrm{e}^{-\int_{x}^{a}\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))},\mathrm{d}z}f_{2}(a;\;\theta,a),\nonumber\end{eqnarray}

which is (11). By conditioning on the first Poisson observation time, for $x\,{\gt}\,a$ , we have

(42) \begin{eqnarray}&&f_{2}(x;\;\theta,a)\nonumber\\&\,=\,&\int_{a}^{\infty}\int_{x\vee y}^{\infty}\mathrm{e}^{-\theta (y-a)}\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi}\right)\nonumber\\&&+\int_{-\infty}^{a}\int_{x}^{\infty}\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge\tau_{\xi(z)}^{-}\right)f_{2}(z;\;\theta,a)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi}\right)\nonumber\\&&+\int_{-\infty}^{a}\int_{x}^{\infty}\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;T_{a}^{+}\,{\lt}\,\tau_{\xi(z)}^{-}\right)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi}\right)\nonumber\\&&-\int_{-\infty}^{a}\int_{x}^{\infty}\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;\tau_{z}^{+}\,{\lt}\, T_{a}^{+}\,{\lt}\,\tau_{\xi(z)}^{-}\right)\nonumber\\&& \times\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi}\right)\!.\end{eqnarray}

By (17) in [Reference Albrecher, Ivanovs and Zhou5] and (10) in [Reference Li and Zhou19], it can be verified that

(43) \begin{eqnarray}\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge\tau_{\xi(z)}^{-}\right)&\,=\,&\mathrm{P}_{y-\xi(z)}\left(\tau_{\overline{\xi}(z)}^{+}\,{\lt}\,T_{a-\xi(z)}^{+}\wedge\tau_{0}^{-}\right)\nonumber\\&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{-\lambda\int_{0}^{\tau_{\overline{\xi}(z)}^{+}}\textbf{1}_{[a-\xi(z), \overline{\xi}(z)]}(X(t))\mathrm{d}t};\;\tau_{\overline{\xi}(z)}^{+}\,{\lt}\,\tau_{0}^{-}\right)\nonumber\\&\,=\,&\frac{W(y-\xi(z))}{W_{a-\xi(z)}^{(0,\lambda)}(\overline{\xi}(z))},\end{eqnarray}

(44) \begin{eqnarray}\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;T_{a}^{+}\,{\lt}\,\tau_{\xi(z)}^{-}\right)&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{-\theta (X(T_{a-\xi(z)}^{+})-(a-\xi(z)))};\;T_{a-\xi(z)}^{+}\,{\lt}\,\tau_{0}^{-}\right)\nonumber\\&\,=\,&\frac{\lambda}{\Phi_{\lambda}+\theta}\frac{W(y-\xi(z))}{Z(a-\xi(z),\Phi_{\lambda})},\end{eqnarray}

for $y\in(\xi(z),a),z\in(x\vee y,\infty)$ , and

(45) \begin{eqnarray}&&\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;\tau_{z}^{+}\,{\lt}\, T_{a}^{+}\,{\lt}\,\tau_{\xi(z)}^{-}\right)\nonumber\\&\,=\,&\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge\tau_{\xi(z)}^{-}\right)\mathrm{E}_{z}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{\xi(z)}^{-}\right)\nonumber\\&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{-\lambda\int_{0}^{\tau_{\overline{\xi}(z)}^{+}}\textbf{1}_{[a-\xi(z), \overline{\xi}(z)]}(X(t))\mathrm{d}t};\;\tau_{\overline{\xi}(z)}^{+}\,{\lt}\,\tau_{0}^{-}\right)\nonumber\\&&\times\mathrm{E}_{\overline{\xi}(z)}\left(\mathrm{e}^{-\theta (X(T_{a-\xi(z)}^{+})-(a-\xi(z)))};\;T_{a-\xi(z)}^{+}\,{\lt}\,\tau_{0}^{-}\right)\nonumber\\&\,=\,&\frac{W(y-\xi(z))}{W_{a-\xi(z)}^{(0,\lambda)}(\overline{\xi}(z))}\,\varphi(\overline{\xi}(z);\;\theta,a-\xi(z)),\quad y\in(\xi(z),a),\ z\in(x\vee y,\infty).\end{eqnarray}

By (6), (42)–(45), we have

(46) \begin{align}&f_{2}(x;\;\theta,a)\nonumber\\&=\lambda W(0)\int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{y}\frac{W_{\lambda}'(\overline{\xi}(w))}{ W_{\lambda}(\overline{\xi}(w))}\,\mathrm{d}w}\mathrm{e}^{-\theta (y-a)}\,\mathrm{d}y\nonumber\\&+\lambda\int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{ W_{\lambda}(\overline{\xi}(w))}\,\mathrm{d}w}\,\mathrm{d}z\int_{a\vee \xi(z)}^{z}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&+\lambda \int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{ W_{\lambda}(\overline{\xi}(w))}\,\mathrm{d}w}f_{2}(z;\;\theta,a)\mathrm{d}z \int_{a\wedge\xi(z)}^{a}\frac{W(y-\xi(z))\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)}{W_{a-\xi(z)}^{(0,\lambda)}(\overline{\xi}(z))}\mathrm{d}y\nonumber\\&+\frac{\lambda^{2}}{\Phi_{\lambda}+\theta}\int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{ W_{\lambda}(\overline{\xi}(w))}\,\mathrm{d}w} \,\mathrm{d}z \int_{a\wedge\xi(z)}^{a}\frac{W(y-\xi(z))\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)}{Z(a-\xi(z),\Phi_{\lambda})}\,\mathrm{d}y\nonumber\\&-\lambda \int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{ W_{\lambda}(\overline{\xi}(w))}\,\mathrm{d}w}\frac{W(y-\xi(z))}{W_{a-\xi(z)}^{(0,\lambda)}(\overline{\xi}(z))}\,\varphi(\overline{\xi}(z);\;\theta,a-\xi(z))\mathrm{d}z\nonumber\\&\times\int_{a\wedge\xi(z)}^{a}\bigg(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\bigg)\mathrm{d}y.\end{align}

Taking derivative on both sides of (46) with respect to x, we get

\begin{eqnarray*}&&f_{2}'(x;\;\theta,a)\nonumber\\&\,=\,&\left(\frac{W_{\lambda}'(\overline{\xi}(x))}{ W_{\lambda}(\overline{\xi}(x))}-\lambda \int_{a\wedge\xi(x)}^{a}\frac{W(y-\xi(x))\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\,\mathrm{d}y\right)f_{2}(x;\;\theta,a)\nonumber\\&&-\lambda W(0)\mathrm{e}^{-\theta (x-a)}-\lambda \int_{a\vee \xi(x)}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&&-\frac{\lambda^{2}}{\Phi_{\lambda}+\theta}\int_{a\wedge\xi(x)}^{a}\frac{W(y-\xi(x))\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)}{Z(a-\xi(x),\Phi_{\lambda})}\,\mathrm{d}y\nonumber\\&&+\lambda\frac{W(y-\xi(x))}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\,\varphi(\overline{\xi}(x);\;\theta,a-\xi(x))\int_{a\wedge\xi(x)}^{a}\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&\,=\,&{\left(\frac{W_{\lambda}'(\overline{\xi}(x))}{ W_{\lambda}(\overline{\xi}(x))}+\frac{W_{\lambda}'(\overline{\xi}(x))}{ W_{\lambda}(\overline{\xi}(x))}\frac{\lambda}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}\frac{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))-W_{\lambda}(\overline{\xi}(x))}{-\lambda}\right.}\nonumber\\&&{\left.-\frac{\lambda}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}\int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y\right)f_2(x;\;\theta,a)}\nonumber\\&&{-\lambda W(0)\mathrm{e}^{-\theta (x-a)}-\lambda \int_{a\vee \xi(x)}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)}\,\mathrm{d}y\nonumber\\\end{eqnarray*}

(47) \begin{eqnarray}&&{-\frac{\lambda^2 \mathrm{e}^{-\theta(\overline{\xi}(x)-a)}}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\int_0^{\overline{\xi}(x)-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y}\nonumber\\&&{\times\int_{a\wedge\xi(x)}^{a}W(y-\xi(x))\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y}\nonumber\\&\,=\,&{\frac{W_{\lambda}'(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}f_2(x;\;\theta,a)}\nonumber\\&&{-\lambda W(0)\mathrm{e}^{-\theta (x-a)}-\lambda \int_{a\vee \xi(x)}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y}\nonumber\\&&{+\lambda \mathrm{e}^{-\theta(\overline{\xi}(x)-a)}\int_0^{\overline{\xi}(x)-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{(W^{(0,\lambda)}_{a-\xi(x)})'(\overline{\xi}(x))}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}-\frac{W'_{\lambda}(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\right)\textbf{1}_{\{a\,{\gt}\,\xi(x)\}}}.\end{eqnarray}

Solving (47) with boundary condition (32) yields

(48) \begin{eqnarray}f_2(x;\;\theta,a)=\mathrm{e}^{\int_a^xh_2(z)\mathrm{d}z}\left(\mathrm{e}^{-\int_a^\infty h_2(z)\mathrm{d}z}\textbf{1}_{\{\theta=0\}}-\int_x^{\infty}g_2(z)\mathrm{e}^{-\int_a^zh_2(w)\mathrm{d}w}\,\mathrm{d}z\right)\!,\end{eqnarray}

where $h_2(x)$ is given by

$$h_2(x)=\frac{W_{\lambda}'(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}.$$

To proceed with the simplification of $f_2$ , we next examine the behavior of $h_2(x)$ as x approaches infinity. We divide our discussion into two distinct cases: (i) ${\xi}(x)\geq a$ and (ii) ${\xi}(x)\,{\lt}\, a$ . Recall that $\lim_{x\rightarrow\infty}\overline{\xi}(x)=\infty$ . In the case ${\xi}(x)\geq a$ , we have

(49) \begin{eqnarray}h_2(x)=\frac{W'_{\lambda}(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))},\end{eqnarray}

which, combined with the facts that $W'_{\lambda}(x)/W_{\lambda}(x)$ is decreasing and $\lim_{x\rightarrow\infty}W'_{\lambda}(x)/W_{\lambda}(x)=\Phi_{\lambda}$ , yields that $h_2(x)\geq \Phi_{\lambda}\,{\gt}\,0$ for all large $x\in\{x;\;\xi(x)\geq a\}$ . In the case $\xi(x)\,{\lt}\,a$ , we have

\begin{eqnarray*}h_2(x)&\,=\,&\frac{W_{\lambda}'(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \nonumber\\&\,=\,&\frac{W_{\lambda}'(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}{W_{\lambda}(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}\nonumber\\[2pt]&\,=\,&\frac{\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}-\lambda \int_{0}^{a-\xi(x)}W(y)\frac{W'_{\lambda}(\overline{\xi}(x)-y)}{W_{\lambda}(\overline{\xi}(x)-y)}\frac{W_{\lambda}(\overline{\xi}(x)-y)}{W_{\lambda}(\overline{\xi}(x))}\mathrm{d}y}{1-\lambda \int_{0}^{a-\xi(x)}W(y)\frac{W_{\lambda}(\overline{\xi}(x)-y)}{W_{\lambda}(\overline{\xi}(x))}\,\mathrm{d}y}\nonumber\\[2pt]&\geq&\frac{\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}-\frac{W'_{\lambda}(x-a)}{W_{\lambda}(x-a)}\lambda \int_{0}^{a-\xi(x)}W(y)\frac{W_{\lambda}(\overline{\xi}(x)-y)}{W_{\lambda}(\overline{\xi}(x))}\,\mathrm{d}y}{1-\lambda \int_{0}^{a-\xi(x)}W(y)\frac{W_{\lambda}(\overline{\xi}(x)-y)}{W_{\lambda}(\overline{\xi}(x))}\,\mathrm{d}y}\nonumber\end{eqnarray*}

(50) \begin{eqnarray}&\,=\,&\frac{\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}-\frac{W_{\lambda}'(x-a)}{W_{\lambda}(x-a)}}{1-\lambda \int_{0}^{a-\xi(x)}W(y)\frac{W_{\lambda}(\overline{\xi}(x)-y)}{W_{\lambda}(\overline{\xi}(x))}\,\mathrm{d}y}+\frac{\frac{W_{\lambda}'(x-a)}{W_{\lambda}(x-a)}\left(1-\lambda \int_{0}^{a-\xi(x)}W(y)\frac{W_{\lambda}(\overline{\xi}(x)-y)}{W_{\lambda}(\overline{\xi}(x))}\,\mathrm{d}y\right)}{1-\lambda \int_{0}^{a-\xi(x)}W(y)\frac{W_{\lambda}(\overline{\xi}(x)-y)}{W_{\lambda}(\overline{\xi}(x))}\,\mathrm{d}y}\nonumber\\&\geq&\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}-\frac{W_{\lambda}'(x-a)}{W_{\lambda}(x-a)}+\frac{W_{\lambda}'(x-a)}{W_{\lambda}(x-a)}\nonumber\\&\,=\,&\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))} \nonumber\\&\rightarrow&\Phi_{\lambda}\,{\gt}\,0,\end{eqnarray}

where we have used the facts that $W'_{\lambda}(x)/W_{\lambda}(x)$ is decreasing in x, $\lim_{x\rightarrow\infty}W'_{\lambda}(x)/W_{\lambda}(x)=\Phi_{\lambda}$ , and $\lim_{x\rightarrow\infty}W_{\lambda}(x+y)/W_{\lambda}(x)=\mathrm{e}^{\Phi_{\lambda}y}$ for any $y\in\mathbb{R}$ . Combining (49) and (50) yields

\begin{align} \mathrm{e}^{-\int_a^\infty h_2(z)\mathrm{d}z}=0,\nonumber\end{align}

which, together with (48), gives (12).

For $x\le a$ , by the Markov property of X, (5), and (38), we have

\begin{eqnarray}f_{3}(x;\;\theta,a)&\;:\!=\;&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,\tau_{a}^{+}\right)+\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{a}^{+}\,{\lt}\,\tau_{\xi}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,\tau_{a}^{+}\right)+\mathrm{P}_{x}\left(\tau_{a}^{+}\,{\lt}\,\tau_{\xi}\right)f_{3}(a;\;\theta,a)\nonumber\\&\,=\,&\int_{x}^{a}\mathrm{e}^{-\int_{x}^{z}\frac{W'(\overline{\xi}(w))}{W(\overline{\xi}(w))}\,\mathrm{d}w}\left(\left({-}\theta+\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\right) Z(\overline{\xi}(z),\theta)+\psi(\theta)W(\overline{\xi}(z))\right)\mathrm{d}z\nonumber\\&&+\mathrm{e}^{-\int_{x}^{a}\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\,\mathrm{d}z}f_{3}(a;\;\theta,a),\nonumber\end{eqnarray}

this gives (13).

Let $e_{\lambda}$ denote the first Poisson observation time. For $x\,{\gt}\,a$ we have

(51) \begin{eqnarray}f_{3}(x;\;\theta,a)&\;:\!=\;&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,T_{a}^{+},\tau_{\xi}\,{\lt}\,e_{\lambda}\right)\nonumber\\&&+\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; e_{\lambda}\,{\lt}\,\tau_{\xi}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{\xi}+\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)}\right)\nonumber\\&&+\int_{-\infty}^{a}\int_{x}^{\infty}\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge\tau_{\xi(z)}^{-}\right)f_{3}(z;\;\theta,a)\nonumber\\&&\hspace{1cm}\times\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi}\right)\nonumber\\&&+\int_{-\infty}^{a}\int_{x}^{\infty}\mathrm{E}_{y}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi(z)}^{-})-\xi(z)\right)};\;\tau_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&&\hspace{1cm}\times\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi}\right)\nonumber\\&&-\int_{-\infty}^{a}\int_{x}^{\infty}\mathrm{E}_{y}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi(z)}^{-})-\xi(z)\right)};\;\tau_{z}^{+}\,{\lt}\,\tau_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&& \times\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi}\right)\!.\end{eqnarray}

By (18) in [Reference Albrecher, Ivanovs and Zhou5] and (10) in [Reference Li and Zhou19], it can be verified that, for $ y\in(\xi(z),a)$ and $z\in(x\vee y,\infty)$ ,

(52) \begin{eqnarray}&&\mathrm{E}_{y}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi(z)}^{-})-\xi(z)\right)};\;\tau_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{\theta X(\tau_{0}^{-})};\;\tau_{0}^{-}\,{\lt}\,T_{a-\xi(z)}^{+}\right)\nonumber\\&\,=\,&Z(y-\xi(z),\theta)-\frac{W(y-\xi(z))}{\theta-\Phi_{\lambda}}\left(\psi(\theta)-\lambda\frac{Z(a-\xi(z),\theta)}{Z(a-\xi(z),\Phi_{\lambda})}\right)\!,\end{eqnarray}

and

(53) \begin{eqnarray}&&\mathrm{E}_{y}\left(\mathrm{e}^{\theta\left(X(\tau_{\xi(z)}^{-})-\xi(z)\right)};\;\tau_{z}^{+}\,{\lt}\,\tau_{\xi(z)}^{-}\,{\lt}\, T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge\tau_{\xi(z)}^{-}\right)\mathrm{E}_{z}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi(z)}^{-})-\xi(z)\right)};\; \tau_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{-\lambda\int_{0}^{\tau_{\overline{\xi}(z)}^{+}}\textbf{1}_{[a-\xi(z), \overline{\xi}(z)]}(X(t))\mathrm{d}t};\;\tau_{\overline{\xi}(z)}^{+}\,{\lt}\,\tau_{0}^{-}\right)\nonumber\\&&\times\mathrm{E}_{\overline{\xi}(z)}\left(\mathrm{e}^{\theta X(\tau_{0}^{-})};\; \tau_{0}^{-}\,{\lt}\,T_{a-\xi(z)}^{+}\right)\nonumber\\&\,=\,&\frac{W(y-\xi(z))}{W_{a-\xi(z)}^{(0,\lambda)}(\overline{\xi}(z))}\phi(\overline{\xi}(z);\;\theta,a-\xi(z)),\end{eqnarray}

and, using a similar compensation formula-based argument as that of (38), we have

(54) \begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{\xi}+\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)}\right)\nonumber\\&\,=\,&\int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{W_{\lambda}(\overline{\xi}(w))}\,\mathrm{d}w}\left(\left({-}\theta+\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}\right) Z_{\lambda}(\overline{\xi}(z),\theta)-(\lambda-\psi(\theta))W_{\lambda}(\overline{\xi}(z))\right)\mathrm{d}z.\end{eqnarray}

From (6), (51)–(54), we have

\begin{eqnarray*}f_{3}(x;\;\theta,a)&\,=\,&\int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{W_{\lambda}(\overline{\xi}(w))}\,\mathrm{d}w}\left(\left({-}\theta+\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}\right) Z_{\lambda}(\overline{\xi}(z),\theta)-(\lambda-\psi(\theta))W_{\lambda}(\overline{\xi}(z))\right)\mathrm{d}z\nonumber\\&&+\lambda\int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{ W_{\lambda}(\overline{\xi}(w))}\mathrm{d}w}f_{3}(z;\;\theta,a)\mathrm{d}z\int_{\xi(z)\wedge a}^{a}\frac{W(y-\xi(z))}{W_{a-\xi(z)}^{(0,\lambda)}(\overline{\xi}(z))}\nonumber\\&&\times\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&&+\lambda\int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{ W_{\lambda}(\overline{\xi}(w))}\, \mathrm{d}w}\,\mathrm{d}z\nonumber\\&&\times\int_{\xi(z)\wedge a}^{a}\left(Z(y-\xi(z),\theta)-\frac{W(y-\xi(z))}{\theta-\Phi_{\lambda}}\left(\psi(\theta)-\lambda\frac{Z(a-\xi(z),\theta)}{Z(a-\xi(z),\Phi_{\lambda})}\right)\right)\nonumber\\&&\times\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\end{eqnarray*}

(55) \begin{eqnarray}&&-\lambda\int_{x}^{\infty}\mathrm{e}^{-\int_{x}^{z}\frac{W_{\lambda}'(\overline{\xi}(w))}{ W_{\lambda}(\overline{\xi}(w))}\,\mathrm{d}w}\,\mathrm{d}z \frac{W(y-\xi(z))}{W_{a-\xi(z)}^{(0,\lambda)}(\overline{\xi}(z))}\nonumber\\[2pt]&&\hspace{0.3cm}\times\int_{\xi(z)\wedge a}^{a}\phi(\overline{\xi}(z);\;\theta,a-\xi(z))\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)\mathrm{d}y.\end{eqnarray}

Differentiating both sides of (55) with respect to x gives

\begin{eqnarray*}&&f_{3}'(x;\;\theta,a)\nonumber\\[2pt]&\,=\,&\left(\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}-\lambda\int_{\xi(x)\wedge a}^{a}\frac{W(y-\xi(x))}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y\right)f_{3}(x;\;\theta,a)\nonumber\\[2pt]&&-\left(\left({-}\theta+\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\right) Z_{\lambda}(\overline{\xi}(x),\theta)-(\lambda-\psi(\theta))W_{\lambda}(\overline{\xi}(x))\right)\nonumber\\[2pt]&&-\lambda \int_{\xi(x)\wedge a}^{a}\left(Z(y-\xi(x),\theta)-\frac{W(y-\xi(x))}{\theta-\Phi_{\lambda}}\left(\psi(\theta)-\lambda\frac{Z(a-\xi(x),\theta)}{Z(a-\xi(x),\Phi_{\lambda})}\right)\right)\nonumber\\[2pt]&&\times\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[2pt]&&+\lambda \frac{W(y-\xi(x))}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\int_{\xi(x)\wedge a}^{a}\phi(\overline{\xi}(x);\;\theta,a-\xi(x))\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[2pt]&\,=\,&{\frac{W_{\lambda}'(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}f_3(x;\;\theta,a)}\nonumber\\[2pt]&&{-\left(\left({-}\theta+\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\right) Z_{\lambda}(\overline{\xi}(x),\theta)-(\lambda-\psi(\theta))W_{\lambda}(\overline{\xi}(x))\right)}\nonumber\\[2pt]&&{-\lambda\int_{\xi(x)\wedge a}^aZ(y-\xi(x),\theta)\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y}\nonumber\\[2pt]&&{+\lambda\int_{\xi(x)\wedge a}^a\frac{W(y-\xi(x))}{\theta-\Phi_{\lambda}}\left(\psi(\theta)-\lambda\frac{Z(a-\xi(x),\theta)}{Z(a-\xi(x),\Phi_{\lambda})}\right)}\nonumber\\[2pt]&&{\times\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y}\nonumber\\[2pt]&&{+\lambda\frac{W(y-\xi(x))}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\int_{\xi(x)\wedge a}^a Z^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x),\theta)\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y}\nonumber\\[2pt]&&{+\frac{\lambda}{\theta-\Phi_{\lambda}}\left(\lambda\frac{Z(a-\xi(x),\theta)}{Z(a-\xi(x),\Phi_{\lambda})}-\psi(\theta)\right)}\nonumber\\[2pt]&&{\times\int_{\xi(x)\wedge a}^aW(y-\xi(x))\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y}\nonumber\end{eqnarray*}

(56) \begin{eqnarray}&\,=\,&{\frac{W_{\lambda}'(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}f_3(x;\;\theta,a)}\nonumber\\[2pt]&&{-\left(\left({-}\theta+\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\right) Z_{\lambda}(\overline{\xi}(x),\theta)-(\lambda-\psi(\theta))W_{\lambda}(\overline{\xi}(x))\right)}\nonumber\\[2pt]&&{-\lambda\int_{\xi(x)\wedge a}^aZ(y-\xi(x),\theta)\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y}\nonumber\\[2pt]&&{+\lambda\frac{Z^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x),\theta)}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\int_{\xi(x)\wedge a}^a W(y-\xi(x))\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y}\nonumber\\[2pt]&\,=\,&{\frac{W_{\lambda}'(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}f_3(x;\;\theta,a)}\nonumber\\[2pt]&&{-\left(\left({-}\theta+\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\right) Z_{\lambda}(\overline{\xi}(x),\theta)-(\lambda-\psi(\theta))W_{\lambda}(\overline{\xi}(x))\right)}\nonumber\\[2pt]&&{-\lambda\int_{\xi(x)\wedge a}^aZ(y-\xi(x),\theta)W_{\lambda}'(x-y)\mathrm{d}y+Z_{\lambda}(\overline{\xi}(x),\theta)\frac{W'_{\lambda}(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\textbf{1}_{\{\xi(x)\leq a\}}}\nonumber\\[2pt]&&+\lambda\frac{Z^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x),\theta)}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\int_{\xi(x)\wedge a}^a W(y-\xi(x))W'_{\lambda}(x-y)\mathrm{d}y\nonumber\\[2pt]&&-\frac{Z^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x),\theta)}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}W'_{\lambda}(\overline{\xi}(x))\textbf{1}_{\{\xi(x)\leq a\}}.\end{eqnarray}

We then obtain (14) by solving (56) with boundary condition (33).

Proof. We provide the proof for $f_2(x;\;\theta,a)$ and $f_3(x;\;\theta,a)$ where $x \in (a, \infty)$ ; the proof for $x \in [0, a]$ is analogous. From Theorem 3.1, we have

\begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{-\theta( X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{\xi}\right)=-\int_{x}^{\infty}g_{2}(z)\,\mathrm{e}^{-\int_{x}^{z}h_{2}(w)\mathrm{d}w}\,\mathrm{d}z,\quad x\in(a,\infty).\nonumber\end{eqnarray}

By the assumption $\xi(z)\equiv\xi(a)$ for all $z\in[a,\infty)$ , we get

\begin{eqnarray}\int_{x}^{z}h_{2}(w)\mathrm{d}w&\,=\,&{\int_x^z\frac{W_{\lambda}'(\overline{\xi}(w))-\lambda \int_{0}^{\overline{\xi}(a)}W(y)W'_{\lambda}(\overline{\xi}(w)-y)\mathrm{d}y}{W^{(0,\lambda)}_{\overline{\xi}(a)}(\overline{\xi}(w))}\,\mathrm{d}w}\nonumber\\[2pt]&\,=\,&{\int_x^z\frac{\mathrm{d}}{\mathrm{d}w}\ln W^{(0,\lambda)}_{\overline{\xi}(a)}(w-\xi(a))\mathrm{d}w}\nonumber\\[2pt]&\,=\,&{\ln\frac{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a))}}.\nonumber\end{eqnarray}

For $x\geq a$ , by definition of $\varphi$ in Section 5, we have

\begin{eqnarray}&&\varphi\left(x-\xi(a);\;\theta,\overline{\xi}(a)\right)=\mathrm{E}_{x}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{\xi(a)}^{-}\right)\nonumber\\&\,=\,&\int_{a}^{\infty}\int_{x\vee y}^{\infty}\mathrm{e}^{-\theta (y-a)}\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi(a)}^{-}\right)\nonumber\\&&+\int_{-\infty}^{a}\int_{x}^{\infty}\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;T_{a}^{+}\,{\lt}\,\tau_{\xi(a)}^{-}\right)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi(a)}^{-}\right)\nonumber\\&\,=\,&\lambda W(0)\int_{x}^{\infty}\frac{W_{\lambda}(x-\xi(a))}{W_{\lambda}(y-\xi(a))}\mathrm{e}^{-\theta (y-a)}\,\mathrm{d}y\nonumber\\&&+\lambda\int_{x}^{\infty}\frac{W_{\lambda}(x-\xi(a))}{W_{\lambda}(z-\xi(a))} \mathrm{d}z\int_{a\vee \xi(z)}^{z}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&&+\frac{\lambda^{2}}{\Phi_{\lambda}+\theta}\int_{x}^{\infty}\frac{W_{\lambda}(x-\xi(a))}{W_{\lambda}(z-\xi(a))} \,\mathrm{d}z \int_{a\wedge\xi(z)}^{a}\frac{W(y-\xi(z))\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)}{Z(a-\xi(z),\Phi_{\lambda})}\,\mathrm{d}y.\nonumber\end{eqnarray}

Differentiate both sides of the above equation and then rearranging terms yields

\begin{eqnarray}&&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)\nonumber\\&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta (z-a)}-\lambda \int_{a\vee \xi(z)}^{z}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&&-\frac{\lambda^{2}}{\Phi_{\lambda}+\theta}\int_{a\wedge\xi(z)}^{a}\frac{W(y-\xi(z))\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)}{Z(a-\xi(z),\Phi_{\lambda})}\,\mathrm{d}y.\nonumber\end{eqnarray}

Hence, $g_{2}$ can be rewritten as

\begin{eqnarray}g_{2}(z)&\,=\,&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)\nonumber\\&&+\lambda\frac{W(y-\xi(z))}{W_{a-\xi(z)}^{(0,\lambda)}(\overline{\xi}(z))}\,\varphi(\overline{\xi}(z);\;\theta,a-\xi(z))\int_{a\wedge\xi(z)}^{a}\left(W_{\lambda}'(z-y)-\frac{W_{\lambda}'(\overline{\xi}(z))}{W_{\lambda}(\overline{\xi}(z))}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&\,=\,&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)+\lambda\frac{\varphi(z-\xi(a);\;\theta,\overline{\xi}(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}\nonumber\\&&\times\int_{0}^{\overline{\xi}(a)}W(y)\left(W_{\lambda}'(z-\xi(a)-y)-\frac{W_{\lambda}'(z-\xi(a))}{W_{\lambda}(z-\xi(a))}W_{\lambda}(z-\xi(a)-y)\right)\mathrm{d}y\nonumber\\&\,=\,&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)\nonumber\\&&+\frac{\varphi(z-\xi(a);\;\theta,\overline{\xi}(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}\left(\frac{W_{\lambda}'(z-\xi(a))}{ W_{\lambda}(z-\xi(a))}W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))-W_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a))\right)\nonumber\end{eqnarray}

\begin{eqnarray*}&\,=\,&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)\nonumber\\[5pt]&&+\varphi(z-\xi(a);\;\theta,\overline{\xi}(a))\left(\frac{W_{\lambda}'(z-\xi(a))}{ W_{\lambda}(z-\xi(a))}-\frac{W_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}\right)\nonumber\\[5pt]&\,=\,&\frac{\mathrm{d}}{\mathrm{d}z}\left(\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)\right)-\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)\frac{W_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}.\end{eqnarray*}

Therefore, we have

\begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,\tau_{\xi}\right)\nonumber\\[5pt]&\,=\,&-\int_{x}^{\infty}g_{2}(z)\,\mathrm{e}^{-\int_{x}^{z}h_{2}(w)\mathrm{d}w}\,\mathrm{d}z=-W_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a))\nonumber\\[5pt]&&\times\int_{x}^{\infty}\left(\frac{\frac{\mathrm{d}}{\mathrm{d}z}\left(\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)\right)}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}-\frac{\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)W_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a))}{\left(W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))\right)^{2}}\right)\mathrm{d}z\nonumber\\[5pt]&\,=\,&-W_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a))\left(\left.\frac{\varphi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}\right|_{z=x}^{\infty}\right)\nonumber\\[5pt]&\,=\,&\varphi\left(x-\xi(a);\;\theta,\overline{\xi}(a)\right)\!,\nonumber\end{eqnarray}

which, together with (25), implies (15).

We then prove the expression for $\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,T_{a}^{+}\right)$ . By Theorem 3.1, we have

\begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,T_{a}^{+}\right)=-\int_{x}^{\infty}g_{3}(z)\,\mathrm{e}^{-\int_{x}^{z}h_{2}(w)\mathrm{d}w}\,\mathrm{d}z,\quad x\in(a,\infty).\nonumber\end{eqnarray}

By the assumption that $\xi(z)\equiv\xi(a)$ for all $z\in[a,\infty)$ , we get

\begin{eqnarray}{\int_{x}^{z}h_{2}(w)\mathrm{d}w=\ln\frac{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a))}.}\nonumber\end{eqnarray}

For $x\geq a$ , from the same arguments as used in (51) we have

\begin{eqnarray}\phi\left(x-\xi(a);\;\theta,\overline{\xi}(a)\right)&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi(a)}^{-})-\xi(a)\right)};\; \tau_{\xi(a)}^{-}\,{\lt}\,T_{a}^{+}\right)=\mathrm{E}_{x}\left(\mathrm{e}^{-\lambda\tau_{\xi(a)}^{-}+\theta \left(X(\tau_{\xi(a)}^{-})-\xi(a)\right)}\right)\nonumber\\[5pt]&&\hspace{-3cm}+\int_{-\infty}^{a}\int_{x}^{\infty}\mathrm{E}_{y}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi(a)}^{-})-\xi(a)\right)};\;\tau_{\xi(a)}^{-}\,{\lt}\,T_{a}^{+}\right)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z, e_{\lambda}\,{\lt}\,\tau_{\xi(a)}^{-}\right)\!,\nonumber\end{eqnarray}

which is equivalent to

\begin{eqnarray}&&\phi\left(x-\xi(a);\;\theta,\overline{\xi}(a)\right)\nonumber\\[1.5pt]&\,=\,&\int_{x}^{\infty}\int_{0}^{\overline{\xi}(a)}\left(Z(y,\theta)-\frac{W(y)}{\theta-\Phi_{\lambda}}\left(\psi(\theta)-\lambda\frac{Z(\overline{\xi}(a),\theta)}{Z(\overline{\xi}(a),\Phi_{\lambda})}\right)\right)\nonumber\\[1.5pt]&&\times\frac{W_{\lambda}(x-\xi(a))}{ W_{\lambda}(z-\xi(a))}\,\lambda\left(W_{\lambda}'(z-\xi(a)-y)-\frac{W_{\lambda}'(z-\xi(a))}{W_{\lambda}(z-\xi(a))}W_{\lambda}(z-\xi(a)-y)\right)\mathrm{d}z \, \mathrm{d}y\nonumber\\[1.5pt]&&+\int_{x}^{\infty}\frac{W_{\lambda}(x-\xi(a))}{W_{\lambda}(z-\xi(a))}\left(\!\left({-}\theta+\frac{W_{\lambda}'(z-\xi(a))}{W_{\lambda}(z-\xi(a))}\right)\! Z_{\lambda}(z-\xi(a),\theta)-(\lambda-\psi(\theta))W_{\lambda}(z-\xi(a))\!\right)\!\mathrm{d}z.\nonumber\end{eqnarray}

Differentiate both sides of the above equation and then rearranging terms yields

\begin{eqnarray}&&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)\nonumber\\[1.5pt]&\,=\,&-\int_{0}^{\overline{\xi}(a)}\left(Z(y,\theta)-\frac{W(y)}{\theta-\Phi_{\lambda}}\left(\psi(\theta)-\lambda\frac{Z(\overline{\xi}(a),\theta)}{Z(\overline{\xi}(a),\Phi_{\lambda})}\right)\right)\nonumber\\[1.5pt]&&\times\,\lambda\left(W_{\lambda}'(z-\xi(a)-y)-\frac{W_{\lambda}'(z-\xi(a))}{W_{\lambda}(z-\xi(a))}W_{\lambda}(z-\xi(a)-y)\right)\mathrm{d}y\nonumber\\[1.5pt]&&-\left(\left({-}\theta+\frac{W_{\lambda}'(z-\xi(a))}{W_{\lambda}(z-\xi(a))}\right) Z_{\lambda}(z-\xi(a),\theta)-(\lambda-\psi(\theta))W_{\lambda}(z-\xi(a))\right)\!.\nonumber\end{eqnarray}

Hence, $g_{3}$ can be rewritten as

\begin{eqnarray}g_{3}(z)&\,=\,&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)+\frac{\phi(z-\xi(a);\;\theta,\overline{\xi}(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}\nonumber\\[1.5pt]&&\times\lambda\int_{0}^{\overline{\xi}(a)}W(y)\left(W_{\lambda}'(z-\xi(a)-y)-\frac{W_{\lambda}'(z-\xi(a))}{W_{\lambda}(z-\xi(a))}W_{\lambda}(z-\xi(a)-y)\right)\mathrm{d}y\nonumber\\[1.5pt]&\,=\,&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)\nonumber\\[1.5pt]&&+\frac{\phi(z-\xi(a);\;\theta,\overline{\xi}(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}\left(\frac{W_{\lambda}'(z-\xi(a))}{ W_{\lambda}(z-\xi(a))}W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))-W_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a))\right)\nonumber\\[1.5pt]&\,=\,&W_{\lambda}(z-\xi(a))\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\lambda}(z-\xi(a))}\right)\nonumber\\[1.5pt]&&+\phi(z-\xi(a);\;\theta,\overline{\xi}(a))\left(\frac{W_{\lambda}'(z-\xi(a))}{ W_{\lambda}(z-\xi(a))}-\frac{W_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}\right)\nonumber\\[1.5pt]&\,=\,&\frac{\mathrm{d}}{\mathrm{d}z}\left(\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)\right)-\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)\frac{W_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a))}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}.\nonumber\end{eqnarray}

Therefore, we have

\begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(\tau_{\xi})-\xi(\overline{X}(\tau_{\xi}))\right)};\; \tau_{\xi}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&-\int_{x}^{\infty}g_{3}(z)\,\mathrm{e}^{-\int_{x}^{z}h_{2}(w)\mathrm{d}w}\,\mathrm{d}z=-W_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a))\nonumber\\&&\times\int_{x}^{\infty}\left(\frac{\frac{\mathrm{d}}{\mathrm{d}z}\left(\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)\right)}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}-\frac{\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)W_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a))}{\left(W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))\right)^{2}}\right)\mathrm{d}z\nonumber\\&\,=\,&-W_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a))\left(\left.\frac{\phi\left(z-\xi(a);\;\theta,\overline{\xi}(a)\right)}{W_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a))}\right|_{z=x}^{\infty}\right)\nonumber\\&\,=\,&\phi\left(x-\xi(a);\;\theta,\overline{\xi}(a)\right)\!,\nonumber\end{eqnarray}

which, together with (22), implies (16).

Proof. Using similar arguments as those adopted in the proof of (31)–(33), one can deduce that

(57) \begin{align} \lim\limits_{x\rightarrow\infty}f_{4}(x,\theta, a)=0, \quad \lim\limits_{x\rightarrow\infty}f_{5}(x,\theta, a)=\textbf{1}_{\{\theta=0\}},\end{align}

where the assumption of $\lim_{z\rightarrow\infty}\overline{\xi}(z)=\infty$ is applied.

For $x\le a$ we have

\begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; \tau_{a}^{+}\,{\lt}\,T_{\xi}\,{\lt}\,T_{a}^{+}\right)+\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,\tau_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{P}_{x}\left(\tau_{a}^{+}\,{\lt}\,T_{\xi}\right)\mathrm{E}_{a}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,T_{a}^{+}\right)+\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,\tau_{a}^{+}\right)\nonumber\\&\,=\,&\left(1-f_{1}(x;\;0,a)\right)\mathrm{E}_{a}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,T_{a}^{+}\right)+f_{1}(x;\;\theta,a),\nonumber\end{eqnarray}

this leads to (17).

For $x\,{\gt}\,a$ , let $e_{\lambda}$ denote the first Poisson observation time, we have

(58) \begin{eqnarray}&&f_{4}(x;\;\theta,a)\;:\!=\;\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; T_{\xi}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\; e_{\lambda}\leq T_{\xi}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\int_{x}^{\infty}\int_{-\infty}^{\xi(z)\wedge a}\mathrm{e}^{\theta \left(y-\xi(z)\right)}\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\nonumber\\&&+\int_{x}^{\infty}\int_{\xi(z)\wedge a}^{a}\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge T_{\xi(z)}^{-}\right)f_{4}(z;\;\theta,a)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\nonumber\\&&+\int_{x}^{\infty}\int_{\xi(z)\wedge a}^{a}\mathrm{E}_{y}\left(\mathrm{e}^{\theta \left(X(T_{\xi(z)}^{-})-\xi(z)\right)};\;T_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\nonumber\\&&-\int_{x}^{\infty}\int_{\xi(z)\wedge a}^{a}\mathrm{E}_{y}\left(\mathrm{e}^{\theta \left(X(T_{\xi(z)}^{-})-\xi(z)\right)};\;\tau_{z}^{+}\,{\lt}\,T_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\!, \end{eqnarray}

where, in the second equality, we have used the fact that

\begin{align} &\mathrm{E}_{x}\left[\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(T_{\xi}))\right)};\;e_{\lambda}\,{\lt}\,T_{\xi}\,{\lt}\,T_{a}^{+}\wedge \ell_{\overline{X}(e_{\lambda})}^{+}\right] \nonumber\\ =& \mathrm{E}_{x}\left[\mathrm{e}^{\theta \left(X(T_{\xi})-\xi(\overline{X}(e_{\lambda}))\right)};\;e_{\lambda}\,{\lt}\,T_{\xi}\,{\lt}\,T_{a}^{+}\wedge \ell_{\overline{X}(e_{\lambda})}^{+}\right] \nonumber\\ =&\mathrm{E}_{x}\left[\mathrm{E}_{x}\left[\left.\mathrm{e}^{\theta \left(X(T_{\xi(\overline{X}(e_{\lambda}))}^{-})-\xi(\overline{X}(e_{\lambda}))\right)};\;e_{\lambda}\,{\lt}\,T_{\xi(\overline{X}(e_{\lambda}))}^{-}\,{\lt}\,T_{a}^{+}\wedge \ell_{\overline{X}(e_{\lambda})}^{+}\right|\mathcal{F}_{e_{\lambda}}\right]\right] \nonumber\\ =& \mathrm{E}_{x}\left[ \left.\left(\textbf{1}_{\{a\,{\gt}\,y\,{\gt}\,\xi(z)\}}\,\mathrm{E}_{y}\left[\mathrm{e}^{\theta \left(X(T_{\xi(z)}^{-})-\xi(z)\right)};\;T_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\wedge \tau_{z}^{+}\right]\right)\right|_{y=X(e_{\lambda}),\,z=\overline{X}(e_{\lambda})}\right] \nonumber\\ =& \mathrm{E}_{x}\left[ \left.\left(\textbf{1}_{\{a\,{\gt}\,y\,{\gt}\,\xi(z)\}}\,\mathrm{E}_{y}\left[\mathrm{e}^{\theta \left(X(T_{\xi(z)}^{-})-\xi(z)\right)};\;T_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right]\right)\right|_{y=X(e_{\lambda}),\,z=\overline{X}(e_{\lambda})}\right] \nonumber\\ &-\mathrm{E}_{x}\left[ \left.\left(\textbf{1}_{\{a\,{\gt}\,y\,{\gt}\,\xi(z)\}}\,\mathrm{E}_{y}\left[\mathrm{e}^{\theta \left(X(T_{\xi(z)}^{-})-\xi(z)\right)};\;\tau_{z}^{+}\,{\lt}\,T_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right]\right)\right|_{y=X(e_{\lambda}),\,z=\overline{X}(e_{\lambda})}\right], \nonumber\end{align}

with $\ell_{b}^{+}\;:\!=\;\inf\{t\geq e_{\lambda};\; X(t)\geq b\}$ . By (19) in [Reference Albrecher, Ivanovs and Zhou5] and (10) in [Reference Li and Zhou19], it can be verified that, for $ y\in(\xi(z),a)$ and $z\in(x\vee y,\infty)$ ,

(59) \begin{eqnarray}&&\mathrm{E}_{y}\left(\mathrm{e}^{\theta \left(X(T_{\xi(z)}^{-})-\xi(z)\right)};\;T_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{\theta X(T_{0}^{-})};\;T_{0}^{-}\,{\lt}\,T_{a-\xi(z)}^{+}\right)\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left(Z(y-\xi(z),\theta)-Z(y-\xi(z),\Phi_{\lambda})\frac{\widetilde{Z}(a-\xi(z),\Phi_{\lambda}, \theta)}{\widetilde{Z}(a-\xi(z),\Phi_{\lambda}, \Phi_{\lambda})}\right)\!,\end{eqnarray}

and

(60) \begin{eqnarray}&&\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge T_{\xi(z)}^{-}\right)\nonumber\\&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{-\lambda\int_{0}^{\tau_{\overline{\xi}(z)}^{+}}\textbf{1}_{[a-\xi(z), \overline{\xi}(z)]\cup ({-}\infty, 0)}(X(t))\mathrm{d}t}\right)\nonumber\\&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{-\lambda\tau_{\overline{\xi}(z)}^{+}+\lambda\int_{0}^{\tau_{\overline{\xi}(z)}^{+}}\textbf{1}_{[0,a-\xi(z)]}(X(t))\mathrm{d}t}\right)\nonumber\\&\,=\,&\rho(y-\xi(z);\;\lambda, a-\xi(z), \overline{\xi}(z)),\end{eqnarray}

and

(61) \begin{eqnarray}&&\mathrm{E}_{y}\left(\mathrm{e}^{\theta \left(X(T_{\xi(z)}^{-})-\xi(z)\right)};\;\tau_{z}^{+}\,{\lt}\,T_{\xi(z)}^{-}\,{\lt}\, T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge T_{\xi(z)}^{-}\right)\mathrm{E}_{z}\left(\mathrm{e}^{\theta \left(X(T_{\xi(z)}^{-})-\xi(\overline{X}(T_{\xi(z)}^{-}))\right)};\; T_{\xi(z)}^{-}\,{\lt}\,T_{a}^{+}\right)\nonumber\\&\,=\,&\mathrm{E}_{y-\xi(z)}\left(\mathrm{e}^{-\lambda\int_{0}^{\tau_{\overline{\xi}(z)}^{+}}\textbf{1}_{[a-\xi(z), \overline{\xi}(z)]\cup ({-}\infty, 0)}(X(t))\mathrm{d}t}\right)\nonumber\\&&\times\mathrm{E}_{\overline{\xi}(z)}\left(\mathrm{e}^{\theta X(T_{0}^{-})};\; T_{0}^{-}\,{\lt}\,T_{a-\xi(z)}^{+}\right)\nonumber\\&\,=\,&\rho(y-\xi(z);\;\lambda, a-\xi(z), \overline{\xi}(z))\upsilon(\overline{\xi}(z);\;\theta,a-\xi(z)).\end{eqnarray}

It follows from (7) and (58)–(61) that

(62) \begin{eqnarray}f_{4}(x;\;\theta,a)&\,=\,&\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}\,\mathrm{d}z\int_{-\infty}^{\xi(z)\wedge a}\mathrm{e}^{\theta \left(y-\xi(z)\right)}\left({-}\Phi_{\lambda}W_{\lambda}(z-y)+W_{\lambda}'(z-y)\right)\mathrm{d}y\nonumber\\&&+\lambda\int_{x}^{\infty}f_{4}(z;\;\theta,a)\mathrm{e}^{-\Phi_{\lambda}(z-x)}\mathrm{d}z\int_{\xi(z)\wedge a}^{a}\rho(y-\xi(z);\;\lambda, a-\xi(z), \overline{\xi}(z))\nonumber\\&&\times\left({-}\Phi_{\lambda}W_{\lambda}(z-y)+W_{\lambda}'(z-y)\right)\mathrm{d}y\nonumber\\&&+\int_{x}^{\infty}\,\mathrm{e}^{-\Phi_{\lambda}(z-x)}\,\mathrm{d}z\int_{\xi(z)\wedge a}^{a}\hspace{-0.1cm}\frac{\lambda\left(Z(y-\xi(z),\theta)-Z(y-\xi(z),\Phi_{\lambda})\frac{\widetilde{Z}(a-\xi(z),\Phi_{\lambda}, \theta)}{\widetilde{Z}(a-\xi(z),\Phi_{\lambda}, \Phi_{\lambda})}\right)}{\lambda-\psi(\theta)}\nonumber\\&&\times\lambda\left({-}\Phi_{\lambda}W_{\lambda}(z-y)+W_{\lambda}'(z-y)\right)\mathrm{d}y\nonumber\\&&-\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}\upsilon(\overline{\xi}(z);\;\theta,a-\xi(z))\nonumber\\&&\times\int_{\xi(z)\wedge a}^{a}\rho(y-\xi(z);\;\lambda, a-\xi(z), \overline{\xi}(z))\nonumber\\&&\times\left({-}\Phi_{\lambda}W_{\lambda}(z-y)+W_{\lambda}'(z-y)\right)\mathrm{d}y.\end{eqnarray}

Taking derivatives with respect to x on both sides of (62) we get

(63) \begin{eqnarray}f_{4}'(x;\;\theta,a)&\,=\,&\left(\Phi_{\lambda}-\lambda\int_{\xi(x)\wedge a}^{a}\rho(y-\xi(x);\;\lambda, a-\xi(x), \overline{\xi}(x))\right.\nonumber\\&&\times\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\Bigg)f_{4}(x;\;\theta,a)\nonumber\\&&-\lambda\int_{-\infty}^{\xi(x)\wedge a}\mathrm{e}^{\theta \left(y-\xi(x)\right)}\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\nonumber\\&&-\,\int_{\xi(x)\wedge a}^{a}\hspace{-0.1cm}\frac{\lambda\left(Z(y-\xi(x),\theta)-Z(y-\xi(x),\Phi_{\lambda})\frac{\widetilde{Z}(a-\xi(x),\Phi_{\lambda}, \theta)}{\widetilde{Z}(a-\xi(x),\Phi_{\lambda}, \Phi_{\lambda})}\right)}{\lambda-\psi(\theta)}\nonumber\\&&\times\lambda\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\nonumber\\&&+\lambda\upsilon(\overline{\xi}(x);\;\theta,a-\xi(x))\int_{\xi(x)\wedge a}^{a}\rho(y-\xi(x);\;\lambda, a-\xi(x), \overline{\xi}(x))\nonumber\\&&\times\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y.\end{eqnarray}

Solving (63) with boundary condition (57) gives (18).

For $x\in[0,a]$ ,

\begin{eqnarray}f_{5}(x;\;\theta,a)&\;:\!=\;&\mathrm{E}_{x}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{\xi}\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{E}_{x}\left(\left.\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; \tau_{a}^{+}\,{\lt}\,T_{a}^{+}\,{\lt}\,T_{\xi}\right|\mathcal{F}_{\tau_{a}^{+}}\right)\right)\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{E}_{a}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{\xi}\right);\; \tau_{a}^{+}\,{\lt}\,T_{\xi}\right)\nonumber\\&\,=\,&\left(1-f_{1}(x;\;0,a)\right)\mathrm{E}_{a}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{\xi}\right)\!,\nonumber\end{eqnarray}

this gives (19).

For $x\in(a,\infty)$ , by conditioning on the first Poisson observation time, we have

(64) \begin{eqnarray}&&f_{5}(x;\;\theta,a)\nonumber\\&\,=\,&\int_{a}^{\infty}\int_{x\vee y}^{\infty}\mathrm{e}^{-\theta (y-a)}\textbf{1}_{\{y\geq\xi(z)\}}\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\nonumber\\&&+\int_{x}^{\infty}\int_{\xi(z)\wedge a}^{a}\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\,T_{a}^{+}\wedge T_{\xi(z)}^{-}\right)f_{5}(z;\;\theta,a)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\nonumber\\&&+\int_{x}^{\infty}\int_{\xi(z)\wedge a}^{a}\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;T_{a}^{+}\,{\lt}\,T_{\xi(z)}^{-}\right)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\nonumber\\&&-\int_{x}^{\infty}\int_{\xi(z)\wedge a}^{a}\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;\tau_{z}^{+}\,{\lt}\, T_{a}^{+}\,{\lt}\,T_{\xi(z)}^{-}\right)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\!.\end{eqnarray}

Hence, by (7), one gets

(65) \begin{eqnarray}&&f_{5}(x;\;\theta,a)\nonumber\\&\,=\,&\lambda W(0)\int_{x}^{\infty}\mathrm{e}^{-\theta (y-a)-\Phi_{\lambda}(y-x)}\,\mathrm{d}y\nonumber\\&&+\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}\, \mathrm{d}z\int_{a\vee \xi(z)}^{z}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&&+\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}f_{5}(z;\;\theta,a)\mathrm{d}z\int_{\xi(z)\wedge a}^{a}\rho(y-\xi(z);\;\lambda, a-\xi(z), \overline{\xi}(z))\nonumber\\&&\times\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&&+\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}\int_{\xi(z)\wedge a}^{a}\frac{\lambda}{\Phi_{\lambda}+\theta}\frac{Z(y-\xi(z),\Phi_{\lambda})}{\widetilde{Z}(a-\xi(z),\Phi_{\lambda},\Phi_{\lambda})}\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&&-\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}\int_{\xi(z)\wedge a}^{a}\rho(y-\xi(z);\;\lambda, a-\xi(z), \overline{\xi}(z))\vartheta(\overline{\xi}(z);\;\theta,a-\xi(z))\nonumber\\&&\times\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y,\end{eqnarray}

where we have also used the following result (see Equation (20) of [Reference Albrecher, Ivanovs and Zhou5])

\begin{eqnarray}\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;T_{a}^{+}\,{\lt}\,\tau_{\xi(z)}^{-}\right)=\frac{\lambda}{\Phi_{\lambda}+\theta}\frac{Z(y-\xi(z),\Phi_{\lambda})}{\widetilde{Z}(a-\xi(z),\Phi_{\lambda},\Phi_{\lambda})},\quad y\leq a,\nonumber\end{eqnarray}

and

\begin{eqnarray}&&\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;\tau_{z}^{+}\,{\lt}\, T_{a}^{+}\,{\lt}\,T_{\xi(z)}^{-}\right)\nonumber\\&\,=\,&\mathrm{P}_{y}\left(\tau_{z}^{+}\,{\lt}\, T_{a}^{+}\wedge T_{\xi(z)}^{-}\right)\mathrm{E}_{z}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{\xi(z)}^{-}\right)\nonumber\\&\,=\,&\rho(y-\xi(z);\;\lambda, a-\xi(z), \overline{\xi}(z))\nonumber\\&&\times\mathrm{E}_{z}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{\xi(z)}^{-}\right)\nonumber\\&\,=\,&\rho(y-\xi(z);\;\lambda, a-\xi(z), \overline{\xi}(z))\nonumber\\&&\times\vartheta(\overline{\xi}(z);\;\theta,a-\xi(z)),\quad y\geq a.\nonumber\end{eqnarray}

Taking derivatives with respect to x on both sides of (65) we get

(66) \begin{eqnarray}&&f_{5}'(x;\;\theta,a)\nonumber\\&\,=\,&\left(\Phi_{\lambda}-\lambda\int_{\xi(x)\wedge a}^{a}\rho(y-\xi(x);\;\lambda, a-\xi(x), \overline{\xi}(x))\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\right)\nonumber\\&&\times f_{5}(x;\;\theta,a)-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\int_{a\vee \xi(x)}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&&-\int_{\xi(x)\wedge a}^{a}\frac{\lambda^{2}}{\Phi_{\lambda}+\theta}\frac{Z(y-\xi(x),\Phi_{\lambda})}{\widetilde{Z}(a-\xi(x),\Phi_{\lambda},\Phi_{\lambda})}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&&+\lambda\int_{\xi(x)\wedge a}^{a}\rho(y-\xi(x);\;\lambda, a-\xi(x), \overline{\xi}(x))\vartheta(\overline{\xi}(x);\;\theta,a-\xi(x))\nonumber\\&&\times\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y.\end{eqnarray}

Solving (66) with boundary condition (57) gives (20).

Proof. We only prove the expression for $f_{5}$ when $\theta\,{\gt}\,0$ . By Theorem 3.2, one has

\begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{-\theta( X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{\xi}\right)=-\int_{x}^{\infty}g_{5}(z)\,\mathrm{e}^{-\int_{x}^{z}h_{4}(w)\mathrm{d}w}\,\mathrm{d}z,\quad x\in(a,\infty).\nonumber\end{eqnarray}

By the assumption that $\xi(z)\equiv\xi(a)$ for all $z\in[a,\infty)$ , we get

\begin{eqnarray}\int_{x}^{z}h_{4}(w)\mathrm{d}w&\,=\,&\int_{x}^{z}\left(\Phi_{\lambda}-\lambda\int_{\xi(a)}^{a}\rho\left(y-\xi(\omega);\;\lambda,a-\xi(\omega),\overline{\xi}(\omega)\right)\right.\nonumber\\&&\left.\times\left(W_{\lambda}'(\omega-y)-\Phi_{\lambda}W_{\lambda}(\omega-y)\right)\mathrm{d}y\right)\mathrm{d}\omega\nonumber\\&\,=\,&(z-x)\Phi_{\lambda}-\int_{x-\xi(a)}^{z-\xi(a)}\frac{I(\omega)}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(\omega-\xi(a),\Phi_{\lambda})}\,\mathrm{d}\omega,\nonumber\end{eqnarray}

where

\begin{eqnarray}&&I(\omega)=\lambda\int_0^{\overline{\xi}(a)}Z(y-\xi(a),\Phi_{\lambda})\left(W_{\lambda}'(\omega-y)-\Phi_{\lambda}W_{\lambda}(\omega-y)\right)\mathrm{d}y.\nonumber\end{eqnarray}

Assuming that $\xi(a)\,{\gt}\,0 $ (since $\xi(a)\leq0$ is trivial), then we separate the interval $[0,\overline{\xi}(a)]$ into two subintervals, $[0,\xi(a)]$ and $(\xi(a),\overline{\xi}(a)].$ Directly from calculations, we obtain

\begin{eqnarray}&&Z_{\overline{\xi}(a)}^{(0,\lambda)'}(\omega-\xi(a),\Phi_{\lambda})-\Phi_{\lambda}Z_{\overline{\xi}(a)}^{(0,\lambda)}(\omega-\xi(a),\Phi_{\lambda})\nonumber\\&\,=\,&\lambda\Phi_{\lambda}\int_{\xi(a)}^{\overline{\xi}(a)}W_{\lambda}(\omega-y)Z(y-\xi(a),\Phi_{\lambda})\mathrm{d}y-\lambda\int_{\xi(a)}^{\overline{\xi}(a)}W_{\lambda}'(\omega-y)Z(y-\xi(a),\Phi_{\lambda})\mathrm{d}y,\nonumber\end{eqnarray}

then

\begin{eqnarray}&&I(\omega)=-\left(Z_{\overline{\xi}(a)}^{(0,\lambda)'}(\omega-\xi(a),\Phi_{\lambda})-\Phi_{\lambda}Z_{\overline{\xi}(a)}^{(0,\lambda)}(\omega-\xi(a),\Phi_{\lambda})\right)+A(\omega),\nonumber\end{eqnarray}

where

\begin{eqnarray}&&A(\omega)=\int_0^{\xi(a)}\frac{\mathrm{e}^{\Phi_{\lambda}(y-\xi(a))}\left(W_{\lambda}'(\omega-y)-\Phi_{\lambda}W_{\lambda}(\omega-y)\right)}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(\omega-\xi(a),\Phi_{\lambda})}\,\mathrm{d}y,\nonumber\end{eqnarray}

by noticing that $Z(y-\xi(a),\Phi_{\lambda})=\mathrm{e}^{\Phi_{\lambda}(y-\xi(a))}$ for $y\leq\xi(a)$ . Next we have

\begin{eqnarray}&&\hspace{-0.6cm}\vartheta(x-\xi(a);\;\theta,\overline{\xi}(a))\nonumber\\&\,=\,&\mathrm{E}_{x}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;T_{a}^{+}\,{\lt}\,T_{\xi(a)}^{-}\right)\nonumber\\&\,=\,&\int_{a}^{\infty}\int_{x\vee y}^{\infty}\mathrm{e}^{-\theta (y-a)}\textbf{1}_{\{y\geq\xi(z)\}}\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\nonumber\\&&+\int_{x}^{\infty}\int_{\xi(z)\wedge a}^{a}\mathrm{E}_{y}\left(\mathrm{e}^{-\theta (X(T_{a}^{+})-a)};\;T_{a}^{+}\,{\lt}\,T_{\xi(a)}^{-}\right)\mathrm{P}_{x}\left(X(e_{\lambda})\in\mathrm{d}y, \overline{X}(e_{\lambda})\in\mathrm{d}z\right)\nonumber\\&\,=\,&\lambda W(0)\int_{x}^{\infty}\mathrm{e}^{-\theta (y-a)-\Phi_{\lambda}(y-x)}\,\mathrm{d}y\nonumber\\&&+\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}\,\mathrm{d}z\int_{a\vee \xi(a)}^{z}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&&+\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}\int_{\xi(a)\wedge a}^{a}\frac{\lambda}{\Phi_{\lambda}+\theta}\frac{Z(y-\xi(z),\Phi_{\lambda})}{\widetilde{Z}(a-\xi(z),\Phi_{\lambda},\Phi_{\lambda})}\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y,\nonumber\end{eqnarray}

differentiating both sides of the above equation and then rearranging terms yields

\begin{eqnarray}&&\frac{\mathrm{d}\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))}{\mathrm{d}z}\nonumber\\&\,=\,&\Phi_{\lambda}\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))-\lambda W(0)\mathrm{e}^{-\theta(z-a)}\nonumber\\&&-\lambda\int_{a\vee \xi(a)}^{z}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&&-\lambda\int_{\xi(a)\wedge a}^{a}\frac{\lambda}{\Phi_{\lambda}+\theta}\frac{Z(y-\xi(z),\Phi_{\lambda})}{\widetilde{Z}(a-\xi(z),\Phi_{\lambda},\Phi_{\lambda})}\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y.\nonumber\end{eqnarray}

Hence, $g_{5}(z)$ can be rewritten as

\begin{eqnarray}g_{5}(z)&\,=\,&\frac{\mathrm{d}\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))}{\mathrm{d}z}-\Phi_{\lambda}\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))\nonumber\\&&-\lambda\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}(z-x)}\int_{\xi(a)\wedge a}^{a}\rho(y-\xi(a);\;\lambda, \overline{\xi}(a), z-\xi(a))\nonumber\\&&\times\vartheta(\overline{\xi}(z);\;\theta,\overline{\xi}(a))\left(W_{\lambda}'(z-y)-\Phi_{\lambda}W_{\lambda}(z-y)\right)\mathrm{d}y\nonumber\\&\,=\,&\frac{\mathrm{d}\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))}{\mathrm{d}z}-\Phi_{\lambda}\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))\nonumber\\&&+\lambda\int_0^{\overline{\xi}(a)}\frac{Z(y-\xi(a),\Phi_{\lambda})}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})}\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))\nonumber\\&&\times\left(W_{\lambda}'(z-y-\xi(a))-\Phi_{\lambda}W_{\lambda}(z-y-\xi(a))\right)\mathrm{d}y.\nonumber\end{eqnarray}

Similarly, separating the interval $[0,\overline{\xi}(a)]$ into two sub-intervals, $[0,\xi(a)]$ and $(\xi(a),\overline{\xi}(a)],$ we obtain

\begin{eqnarray}&&g_5(z)\nonumber\\&\,=\,&\vartheta'(z-\xi(a);\;\theta,\overline{\xi}(a))-\Phi_{\lambda}\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))\nonumber\\&&+\vartheta(\overline{\xi}(z);\;\theta,\overline{\xi}(a))\lambda\int_0^{\xi(a)}\frac{\left(W_{\lambda}'(z-y-\xi(a))-\Phi_{\lambda}W_{\lambda}(z-y-\xi(a))\right)\mathrm{e}^{\Phi_{\lambda}(y-\xi(a))}}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})}\,\mathrm{d}y\nonumber\\&&-\vartheta(\overline{\xi}(z);\;\theta,\overline{\xi}(a))\frac{(Z_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a),\Phi_{\lambda})-\Phi_{\lambda}Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda}))}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})}\nonumber\\&\,=\,&\vartheta'(z-\xi(a);\;\theta,\overline{\xi}(a))-\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))\frac{Z_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a),\Phi_{\lambda})}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})}+\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))B(z),\nonumber\end{eqnarray}

where

\begin{eqnarray}&&B(z)=\int_0^{\xi(a)}\frac{\mathrm{e}^{\Phi_{\lambda}(y-\xi(a))}\left(W_{\lambda}'(z-y-\xi(a))-\Phi_{\lambda}W_{\lambda}(z-y-\xi(a))\right)}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})}\,\mathrm{d}y.\nonumber\end{eqnarray}

Therefore, we have

\begin{eqnarray}&&\mathrm{E}_{x}\left(\mathrm{e}^{-\theta( X(T_{a}^{+})-a)};\; T_{a}^{+}\,{\lt}\,T_{\xi}\right)=-\int_{x}^{\infty}g_5(z)\,\mathrm{e}^{-\int_{x}^{z}h_4(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\&\,=\,&-Z_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a),\Phi_{\lambda})\int_x^{\infty}\left(\frac{\vartheta(\overline{\xi}(z);\;\theta,\overline{\xi}(a))A(z)}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})\mathrm{e}^{-\int_x^z A(\omega)\mathrm{d}w}}\right.\nonumber\\&&\left.+\frac{\vartheta'(z-\xi(a);\;\theta,\overline{\xi}(a))}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})\mathrm{e}^{-\int_x^z A(\omega)\mathrm{d}w}}-\frac{\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))Z_{\overline{\xi}(a)}^{(0,\lambda)'}(z-\xi(a),\Phi_{\lambda})}{\left(Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})\right)^2\mathrm{e}^{-\int_x^z A(\omega)\mathrm{d}w}}\right)\mathrm{d}z\nonumber\\&\,=\,&-Z_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a),\Phi_{\lambda})\int_x^{\infty}\left(\frac{\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})\mathrm{e}^{-\int_x^z A(\omega)\mathrm{d}w}}\right)'_z \,\mathrm{d}z\nonumber\\&\,=\,&-Z_{\overline{\xi}(a)}^{(0,\lambda)}(x-\xi(a),\Phi_{\lambda})\left(\left.\frac{\vartheta(z-\xi(a);\;\theta,\overline{\xi}(a))}{Z_{\overline{\xi}(a)}^{(0,\lambda)}(z-\xi(a),\Phi_{\lambda})\mathrm{e}^{-\int_x^z A(\omega)\mathrm{d}w}}\right|_{z=x}^{\infty}\right)\nonumber\\&\,=\,&\vartheta(x-\xi(a);\;\theta,\overline{\xi}(a)).\nonumber\end{eqnarray}

This completes the proof.

Proof. We recall from Theorem 3.1 that

(67) \begin{align} f_{1}(x;\;\theta,a) =& \frac{\lambda}{\lambda-\psi(\theta)}\int_x^{a}\left(\frac{Z'(\overline{\xi}(z),\Phi_{\lambda})}{Z(\overline{\xi}(z), \Phi_{\lambda})}Z(\overline{\xi}(z),\theta)-Z^{\prime}(\overline{\xi}(z),\theta)\right)\mathrm{e}^{-\int_x^z\frac{Z' (\overline{\xi}(w),\Phi_{\lambda})}{Z(\overline{\xi}(w),\Phi_{\lambda})}\,\mathrm{d}w},\mathrm{d}z. \end{align}

Note that for large $\lambda\,{\gt}\,0$ , we have

(68) \begin{align} \frac{Z'(\overline{\xi}(x),\Phi_{\lambda})}{Z(\overline{\xi}(x),\Phi_{\lambda})}=&\frac{\Phi_{\lambda}Z(\overline{\xi}(x),\Phi_{\lambda})-\lambda W(\overline{\xi}(x))}{Z(\overline{\xi}(x),\Phi_{\lambda})}.\nonumber\\=&\Phi_{\lambda}-\frac{\lambda W(\overline{\xi}(x))}{\mathrm{e}^{\Phi_{\lambda}\overline{\xi}(x)}\left(1-\lambda\int_0^{\overline{\xi}(x)}\mathrm{e}^{-\Phi_{\lambda}z}W(z)\mathrm{d}z\right)}\nonumber\\=&\Phi_{\lambda}-\frac{W(\overline{\xi}(x))}{\mathrm{e}^{\Phi_{\lambda}\overline{\xi}(x)}\int_{\overline{\xi}(x)}^{\infty}\mathrm{e}^{-\Phi_{\lambda}z}W(z),\mathrm{d}z}\nonumber\\=&\Phi_{\lambda}-\frac{\Phi_{\lambda} W(\overline{\xi}(x))}{\int_0^{\infty}\Phi_{\lambda}\mathrm{e}^{-\Phi_{\lambda}u}W(\overline{\xi}(x)+u)\mathrm{d}u}\nonumber\\=&\Phi_{\lambda}-\frac{\Phi_{\lambda} W(\overline{\xi}(x))}{\int_0^{\infty}\mathrm{e}^{-z}W(\overline{\xi}(x)+\frac{z}{\Phi_{\lambda}})\mathrm{d}z}\nonumber\\=&\Phi_{\lambda}\frac{\mathrm{E}\left(W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})-W(\overline{\xi}(x))\right)}{\mathrm{E}\left(W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})\right)}\quad (e_{1}\sim \exp(1) \text{ is a random variable})\nonumber\\=&\frac{\mathrm{E}\left(W^{\prime}(\overline{\xi}(x)+\theta\frac{e_{1}}{\Phi_{\lambda}})\,e_{1}\right)}{\mathrm{E}\left(W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})\right)}\quad (\theta\in(0,1))\nonumber\\=&\frac{\mathrm{E}\left(\frac{W^{\prime}(\overline{\xi}(x)+\theta\frac{e_{1}}{\Phi_{\lambda}})}{W(\overline{\xi}(x)+\theta\frac{e_{1}}{\Phi_{\lambda}})}W(\overline{\xi}(x)+\theta\frac{e_{1}}{\Phi_{\lambda}})e_{1}\right)}{\mathrm{E}\left(W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})\right)}.\end{align}

Utilizing (68) and recalling the facts that $x\mapsto \frac{W^{\prime}(x)}{W(x)}$ is decreasing and $x\mapsto W(x)$ is increasing on $(0,\infty)$ , we have

\begin{align}\frac{\mathrm{E}\left(\frac{W^{\prime}(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})}{W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})}W(\overline{\xi}(x))e_{1}\right)}{\mathrm{E}\left(W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})\right)} \leq \frac{Z'(\overline{\xi}(x),\Phi_{\lambda})}{Z(\overline{\xi}(x),\Phi_{\lambda})} \leq \frac{\mathrm{E}\left(\frac{W^{\prime}(\overline{\xi}(x))}{W(\overline{\xi}(x))}W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})e_{1}\right)}{\mathrm{E}\left(W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})\right)},\nonumber\end{align}

which, together with the monotone convergence theorem, yields

\begin{align} \frac{W^{\prime}(\overline{\xi}(x))}{W(\overline{\xi}(x))}=&\frac{\mathrm{E}\left(\frac{W^{\prime}(\overline{\xi}(x))}{W(\overline{\xi}(x))}W(\overline{\xi}(x))e_{1}\right)}{W(\overline{\xi}(x))}\nonumber\\=&\lim\limits_{\lambda\rightarrow\infty}\frac{\mathrm{E}\left(\frac{W^{\prime}(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})}{W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})}W(\overline{\xi}(x))e_{1}\right)}{\mathrm{E}\left(W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})\right)} \nonumber\\ \leq & \lim\limits_{\lambda\rightarrow\infty} \frac{Z'(\overline{\xi}(x),\Phi_{\lambda})}{Z(\overline{\xi}(x),\Phi_{\lambda})} \nonumber\\ \leq & \lim\limits_{\lambda\rightarrow\infty}\frac{\mathrm{E}\left(\frac{W^{\prime}(\overline{\xi}(x))}{W(\overline{\xi}(x))}W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})e_{1}\right)}{\mathrm{E}\left(W(\overline{\xi}(x)+\frac{e_{1}}{\Phi_{\lambda}})\right)} = \frac{\mathrm{E}\left(\frac{W^{\prime}(\overline{\xi}(x))}{W(\overline{\xi}(x))}W(\overline{\xi}(x))e_{1}\right)}{W(\overline{\xi}(x))}=\frac{W^{\prime}(\overline{\xi}(x))}{W(\overline{\xi}(x))}.\nonumber\end{align}

What follows is the standard application of the dominated convergence theorem to (67) in order to derive (21). We mention that identifying the dominating function is not difficult, since the integrand of (67) is continuous on the closed interval [x, a], and the function ${Z'(\overline{\xi}(x),\Phi_{\lambda})}/{Z(\overline{\xi}(x),\Phi_{\lambda})}$ converges to 0 as $x\rightarrow\infty$ , i.e.

\begin{align} &\lim\limits_{x\rightarrow\infty} \frac{Z'(x,\Phi_{\lambda})}{Z(x,\Phi_{\lambda})}=\lim\limits_{x\rightarrow\infty}\frac{\lambda\mathrm{e}^{\Phi_{\lambda}x}\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}y}W^{\prime}(y)\mathrm{d}y}{\lambda\mathrm{e}^{\Phi_{\lambda}x}\int_{x}^{\infty}\mathrm{e}^{-\Phi_{\lambda}y}W(y)\mathrm{d}y}=\lim\limits_{x\rightarrow\infty}\frac{W^{\prime}(x)}{W(x)}=0, \quad \lambda\in(0,\infty).\nonumber\end{align}

The proof is complete.

Proof. From Theorem 3.1 it follows that

\begin{eqnarray}h_1(x)&\,=\,&\Phi_{\lambda} -\lambda \frac{W((1-k)x+d)}{Z((1-k)x+d,\Phi_{\lambda})}\nonumber\\[5pt]&\,=\,&\frac{Z'((1-k)x+d,\Phi_{\lambda})}{Z((1-k)x+d,\Phi_{\lambda})}=\frac{1}{1-k}\frac{\mathrm{d}}{\mathrm{d}x}\ln{Z((1-k)x+d,\Phi_{\lambda})},\nonumber\end{eqnarray}

and, hence, we have

(69) \begin{eqnarray}\mathrm{e}^{-\int_0^x h_1(z)\mathrm{d}z}=\mathrm{e}^{-\frac{1}{1-k}\int_0^x\frac{\mathrm{d}}{\mathrm{d}z}\ln{Z((1-k)z+d,\Phi_{\lambda})}\,\mathrm{d}z}=\left(\frac{Z((1-k)x+d,\Phi_{\lambda})}{Z(d,\Phi_{\lambda})}\right)^{-\frac{1}{1-k}}.\end{eqnarray}

In addition, we also have

\begin{eqnarray}g_1(x)&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left({-}\theta Z(\overline{\xi}(x),\theta)+\psi(\theta)W(\overline{\xi}(x))\right)\nonumber\\[5pt]&&-\frac{\lambda}{\lambda-\psi(\theta)}\frac{Z(\overline{\xi}(x),\theta)}{Z(\overline{\xi}(x),\Phi_{\lambda})}\left({-}\Phi_{\lambda} Z(\overline{\xi}(x),\Phi_{\lambda})+\lambda W(\overline{\xi}(x))\right)\nonumber\\[5pt]&\,=\,&-\frac{\lambda}{\lambda-\psi(\theta)}Z'(\overline{\xi}(x),\theta)+\frac{\lambda}{\lambda-\psi(\theta)}\frac{Z'(\overline{\xi}(x),\Phi_{\lambda})}{Z(\overline{\xi}(x),\Phi_{\lambda})}Z(\overline{\xi}(x),\theta)\nonumber\\[5pt]&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left(\frac{Z'((1-k)x+d,\Phi_{\lambda})}{Z((1-k)x+d,\Phi_{\lambda})}Z((1-k)x+d,\theta)-Z'((1-k)x+d,\theta)\right)\nonumber\\[5pt]&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left({Z((1-k)x+d,\theta)}h_1(x)-Z'((1-k)x+d,\theta)\right)\!,\nonumber\end{eqnarray}

which together with (10) and (69) yields

\begin{eqnarray*}&&f_1(x;\;\theta,a)\nonumber\\&\,=\,&\frac{\lambda \mathrm{e}^{\int_{0}^{x}h_{1}(z)\mathrm{d}z}}{\lambda-\psi(\theta)}\int_x^a\left({Z((1-k)z+d,\theta)}h_1(z)-Z'((1-k)z+d,\theta)\right)\mathrm{e}^{-\int_{0}^{z}h_{1}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\&\,=\,&\frac{\lambda \mathrm{e}^{\int_{0}^{x}h_{1}(z)\mathrm{d}z}}{\lambda-\psi(\theta)}\int_x^a\left({Z((1-k)z+d,\theta)}h_1(z)-(1-k)Z'((1-k)z+d,\theta)\right)\mathrm{e}^{-\int_{0}^{z}h_{1}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\&&-\frac{\lambda k\mathrm{e}^{\,\int_{0}^{x}h_{1}(z)\mathrm{d}z}}{\lambda-\psi(\theta)}\int_x^a Z'((1-k)z+d,\theta)\mathrm{e}^{-\int_{0}^{z}h_{1}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\end{eqnarray*}

\begin{eqnarray*}&\,=\,&-\frac{\lambda \mathrm{e}^{\,\int_{0}^{x}h_{1}(z)\mathrm{d}z}}{\lambda-\psi(\theta)}\int_x^a\frac{\mathrm{d}}{\mathrm{d}z}\left[Z((1-k)z+d,\theta)\mathrm{e}^{-\int_{0}^{z}h_{1}(w)\mathrm{d}w}\right]\,\mathrm{d}z\nonumber\\&&-\frac{\lambda k}{\lambda-\psi(\theta)}\int_x^a Z'((1-k)z+d,\theta)\mathrm{e}^{-\int_{x}^{z}h_{1}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left[Z((1-k)x+d,\theta)-Z((1-k)a+d,\theta)\mathrm{e}^{-\int_{x}^{a}h_{1}(z)\mathrm{d}z}\right]\nonumber\\&&-\frac{\lambda k}{\lambda-\psi(\theta)}\int_x^a Z'((1-k)z+d,\theta)\mathrm{e}^{-\int_{x}^{z}h_{1}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left[Z((1-k)x+d,\theta)-Z((1-k)a+d,\theta)\left(\frac{Z((1-k)a+d,\Phi_{\lambda})}{Z((1-k)x+d,\Phi_{\lambda})}\right)^{-\frac{1}{1-k}}\right]\nonumber\\&&-\frac{\lambda k}{\lambda-\psi(\theta)}\int_x^a Z'((1-k)z+d,\theta)\left(\frac{Z((1-k)z+d,\Phi_{\lambda})}{Z((1-k)x+d,\Phi_{\lambda})}\right)^{-\frac{1}{1-k}}\,\mathrm{d}z.\nonumber \end{eqnarray*}

The proof is complete.

Proof. From Theorem 3.1, we have

\begin{eqnarray}h_1(x)=\Phi_{\lambda}-\frac{\lambda W(\overline{\xi}(x))}{Z(\overline{\xi}(x),\Phi_{\lambda})}=\Phi_{\lambda} -\lambda \frac{W(x+d)}{Z(x+d,\Phi_{\lambda})}=\frac{Z'(x+d,\Phi_{\lambda})}{Z(x+d,\Phi_{\lambda})}=\frac{\mathrm{d}}{\mathrm{d}x}\ln{Z(x+d,\Phi_{\lambda})},\nonumber\end{eqnarray}

and, hence, we have

\begin{eqnarray}\mathrm{e}^{-\int_0^x h_1(z)\mathrm{d}z}=\mathrm{e}^{-\int_0^x\frac{\mathrm{d}}{\mathrm{d}z}\ln{Z(z+d,\Phi_{\lambda})}\,\mathrm{d}x}=\frac{Z(x+d,\Phi_{\lambda})}{Z(d,\Phi_{\lambda})}.\nonumber\end{eqnarray}

In addition, we also have

\begin{eqnarray}g_1(x)&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left({-}\theta Z(\overline{\xi}(x),\theta)+\psi(\theta)W(\overline{\xi}(x))\right)\nonumber\\&&-\frac{\lambda}{\lambda-\psi(\theta)}\frac{Z(\overline{\xi}(x),\theta)}{Z(\overline{\xi}(x),\Phi_{\lambda})}\left({-}\Phi_{\lambda} Z(\overline{\xi}(x),\Phi_{\lambda})+\lambda W(\overline{\xi}(x))\right)\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left(\frac{Z'(x+d,\Phi_{\lambda})}{Z(x+d,\Phi_{\lambda})}Z(x+d,\theta)-Z'(x+d,\theta)\right)\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left({Z(x+d,\theta)}h_1(x)-Z'(x+d,\theta)\right)\!,\nonumber\end{eqnarray}

which together with (10) and (69) yields

\begin{eqnarray}f_1(x;\;\theta,a)&\,=\,&\frac{\lambda \mathrm{e}^{\,\int_{0}^{x}h_{1}(z)\mathrm{d}z}}{\lambda-\psi(\theta)}\int_x^a\left({Z(z+d,\theta)}h_1(z)-Z'(z+d,\theta)\right)\mathrm{e}^{-\int_{0}^{z}h_{1}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\&\,=\,&-\frac{\lambda \mathrm{e}^{\,\int_{0}^{x}h_{1}(z)\mathrm{d}z}}{\lambda-\psi(\theta)}\int_x^a\frac{\mathrm{d}}{\mathrm{d}z}\left[Z(z+d,\theta)\mathrm{e}^{-\int_{0}^{z}h_{1}(w)\mathrm{d}w}\right]\,\mathrm{d}z\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left[Z(x+d,\theta)-Z(a+d,\theta)\mathrm{e}^{-\,\int_{x}^{a}h_{1}(z)\mathrm{d}z}\right]\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}\left[Z(x+d,\theta)-Z(a+d,\theta)\frac{Z(x+d,\Phi_{\lambda})}{Z(a+d,\Phi_{\lambda})}\right].\nonumber\end{eqnarray}

Furthermore, from (9) and (70), it follows that

\begin{eqnarray}f_1(x;\;\theta,\infty)&\,=\,&\lim_{a\rightarrow\infty}f_1(x;\;\theta,a)\nonumber\\&\,=\,&\lim_{a\rightarrow\infty}\frac{\lambda}{\lambda-\psi(\theta)}\left[Z(x+d,\theta)-Z(a+d,\theta)\frac{Z(x+d,\Phi_{\lambda})}{Z(a+d,\Phi_{\lambda})}\right]\nonumber\\&\,=\,&\frac{\lambda}{\lambda-\psi(\theta)}Z(x+d,\theta)-Z(x+d,\Phi_{\lambda})\frac{\psi(\theta)(\Phi_{\lambda}-\Phi)}{(\lambda-\psi(\theta))(\theta-\Phi)}.\nonumber\end{eqnarray}

From Theorem 3.1, we have

\begin{eqnarray}h_2(x)&\,=\,&\frac{W_{\lambda}'(\overline{\xi}(x))-\lambda \int_{0}^{a-\xi(x)}W(y)W'_{\lambda}(\overline{\xi}(x)-y)\mathrm{d}y}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}\nonumber\\&\,=\,&\frac{(W^{(0,\lambda)}_{a-\xi(x)})'(\overline{\xi}(x))}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}=\frac{\mathrm{d}}{\mathrm{d}x}\ln{W^{(0,\lambda)}_{a+d}(x+d)},\nonumber\end{eqnarray}

and, hence, we have

(70) \begin{eqnarray}\mathrm{e}^{-\int_a^xh_2(z)\mathrm{d}z}=\frac{W^{(0,\lambda)}_{a+d}(a+d)}{W^{(0,\lambda)}_{a+d}(x+d)}.\end{eqnarray}

In addition, we also have

(71) \begin{eqnarray}g_2(x)&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta (x-a)}-\lambda \int_{a\vee \xi(x)}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&&+\lambda \mathrm{e}^{-\theta(\overline{\xi}(x)-a)}\int_0^{\overline{\xi}(x)-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{(W^{(0,\lambda)}_{a-\xi(x)})'(\overline{\xi}(x))}{W^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x))}-\frac{W'_{\lambda}(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\right)\textbf{1}_{\{a\,{\gt}\,\xi(x)\}}\nonumber\\&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\int_{a}^x\mathrm{e}^{-\theta(y-a)}\left(W'_{\lambda}(x-y)-\frac{W'_{\lambda}(x+d)}{W_{\lambda}(x+d)}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&&+\lambda \mathrm{e}^{-\theta(x+d-a)}\int_0^{x+d-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{(W^{(0,\lambda)}_{a+d})'(x+d)}{W^{(0,\lambda)}_{a+d}(x+d)}-\frac{W'_{\lambda}(x+d)}{W_{\lambda}(x+d)}\right)\nonumber\\&\,=\,&-\lambda W_{\lambda}(x-a)+\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{W'_{\lambda}(x+d)}{W_{\lambda}(x+d)}+\theta\right)\nonumber\\&&+\lambda \mathrm{e}^{-\theta(x+d-a)}\int_0^{x+d-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{(W^{(0,\lambda)}_{a+d})'(x+d)}{W^{(0,\lambda)}_{a+d}(x+d)}-\frac{W'_{\lambda}(x+d)}{W_{\lambda}(x+d)}\right)\!.\end{eqnarray}

It follows from (70) and (71) that, for $x\in( a,\infty)$ ,

\begin{eqnarray}&&f_2(x;\;\theta,a)\nonumber\\&\,=\,&-\int_{x}^{\infty}g_2(z)\mathrm{e}^{-\int_x^zh_2(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\&\,=\,&\int_x^{\infty}\left\{\lambda W_{\lambda}(z-a)-\lambda \mathrm{e}^{-\theta(z-a)}\int_0^{z-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{W'_{\lambda}(z+d)}{W_{\lambda}(z+d)}+\theta\right)\right.\nonumber\\&&\left.-\lambda \mathrm{e}^{-\theta(z+d-a)}\int_0^{z+d-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{(W^{(0,\lambda)}_{a+d})'(z+d)}{W^{(0,\lambda)}_{a+d}(z+d)}-\frac{W'_{\lambda}(z+d)}{W_{\lambda}(z+d)}\right)\right\}\frac{W^{(0,\lambda)}_{a+d}(x+d)}{W^{(0,\lambda)}_{a+d}(z+d)}\,\mathrm{d}z.\nonumber\end{eqnarray}

In addition, it follows from (11) that, for $x\in[0,a]$ ,

(72) \begin{eqnarray}&&f_2(x;\;\theta,a)\nonumber\\[5pt]&\,=\,&\mathrm{e}^{-\int_x^a\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}}\,\mathrm{d}zf(a;\;\theta,a)\nonumber\\[4pt]&\,=\,&\frac{W(x+d)}{W(a+d)}\int_a^{\infty}\left\{\lambda W_{\lambda}(z-a)-\lambda \mathrm{e}^{-\theta(z-a)}\int_0^{z-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{W'_{\lambda}(z+d)}{W_{\lambda}(z+d)}+\theta\right)\right.\nonumber\\[4pt]&&\left.-\lambda \mathrm{e}^{-\theta(z+d-a)}\int_0^{z+d-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\left(\frac{(W^{(0,\lambda)}_{a+d})'(z+d)}{W^{(0,\lambda)}_{a+d}(z+d)}-\frac{W'_{\lambda}(z+d)}{W_{\lambda}(z+d)}\right)\right\}\frac{W^{(0,\lambda)}_{a+d}(a+d)}{W^{(0,\lambda)}_{a+d}(z+d)}\,\mathrm{d}z.\nonumber \\&&\end{eqnarray}

Furthermore, from Theorem 3.1, we have

(73) \begin{eqnarray}&&g_3(x)\nonumber\\[4pt]&\,=\,&-\left(\left({-}\theta+\frac{W_{\lambda}'(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\right) Z_{\lambda}(\overline{\xi}(x),\theta)-(\lambda-\psi(\theta))W_{\lambda}(\overline{\xi}(x))\right)\nonumber\\[4pt]&&-\lambda\int_{\xi(x)\wedge a}^aZ(y-\xi(x),\theta)W_{\lambda}'(x-y)\mathrm{d}y+Z_{\lambda}(\overline{\xi}(x),\theta)\frac{W'_{\lambda}(\overline{\xi}(x))}{W_{\lambda}(\overline{\xi}(x))}\textbf{1}_{\{\xi(x)\leq a\}}\nonumber\\[4pt]&&+\lambda\frac{Z^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x),\theta)}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}\int_{\xi(x)\wedge a}^a W(y-\xi(x))W'_{\lambda}(x-y)\mathrm{d}y-\frac{Z^{(0,\lambda)}_{a-\xi(x)}(\overline{\xi}(x),\theta)}{W_{a-\xi(x)}^{(0,\lambda)}(\overline{\xi}(x))}W'_{\lambda}(\overline{\xi}(x))\textbf{1}_{\{\xi(x)\leq a\}}\nonumber\\[4pt]&\,=\,&-\left(Z_{\lambda}(x+d,\theta)\frac{W'_{\lambda}(x+d)}{W_{\lambda}(x+d)}-Z'_{\lambda}(x+d,\theta)\right)\nonumber\\[4pt]&&-\lambda\int_0^{a+d}Z(y,\theta)W'_{\lambda}(x+d-y)\mathrm{d}y+Z_{\lambda}(x+d,\theta)\frac{W'_{\lambda}(x+d)}{W_{\lambda}(x+d)}\nonumber\\[5pt]&&+\lambda\frac{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}{W_{a+d}^{(0,\lambda)}(x+d)}\int_0^{a+d}W(y)W'_{\lambda}(x+d-y)\mathrm{d}y-\frac{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}{W_{a+d}^{(0,\lambda)}(x+d)}W'_{\lambda}(x+d)\nonumber\\[5pt]&\,=\,&(Z^{(0,\lambda)}_{a+d})'(x+d,\theta)-\frac{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}{W_{a+d}^{(0,\lambda)}(x+d)}(W_{a+d}^{(0,\lambda)})'(x+d).\end{eqnarray}

It follows from (14) and (73) yields that for $x\in(a,\infty)$

\begin{eqnarray}f_3(x;\;\theta,a)&\,=\,&-\int_{x}^{\infty}g_{3}(z)\,\mathrm{e}^{-\int_{x}^{z}h_{2}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\[5pt]&\,=\,&-\int_x^{\infty}\left((Z^{(0,\lambda)}_{a+d})'(z+d,\theta)-Z^{(0,\lambda)}_{a+d}(z+d,\theta)h_2(z)\right)\mathrm{e}^{-\int_{x}^{z}h_{2}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\[5pt]&\,=\,&-\int_x^{\infty}\frac{\mathrm{d}}{\mathrm{d}z}\left(Z^{(0,\lambda)}_{a+d}(z+d,\theta)\mathrm{e}^{-\int_{x}^{z}h_{2}(w)\mathrm{d}w}\right)\mathrm{d}z\nonumber\\[5pt]&\,=\,&Z^{(0,\lambda)}_{a+d}(x+d,\theta)-W^{(0,\lambda)}_{a+d}(x+d)\left(\frac{\psi(\theta)}{\theta-\Phi_{\lambda}}-\frac{\lambda}{\theta-\Phi_{\lambda}}\frac{Z(a+d,\theta)}{Z(a+d,\Phi_{\lambda})}\right)\!.\nonumber\end{eqnarray}

Then, it follows from (13) that for $x\in[0,a]$ ,

\begin{eqnarray}f_{3}(x;\;\theta,a)&\,=\,&\mathrm{e}^{-\int_{x}^{a}\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\,\mathrm{d}z}f_{3}(a;\;\theta,a)+\int_{x}^{a}\mathrm{e}^{\theta \xi(z)}\mathrm{e}^{-\int_{x}^{z}\frac{W'(\overline{\xi}(w))}{W(\overline{\xi}(w))}\,\mathrm{d}w}\nonumber\\&&\times\left(\left({-}\theta+\frac{W'(\overline{\xi}(z))}{W(\overline{\xi}(z))}\right) Z(\overline{\xi}(z),\theta)+\psi(\theta)W(\overline{\xi}(z))\right)\mathrm{d}z\nonumber\\&\,=\,&\frac{W(x+d)}{W(a+d)}\left(Z^{(0,\lambda)}_{a+d}(a+d,\theta)-W^{(0,\lambda)}_{a+d}(a+d)\left(\frac{\psi(\theta)}{\theta-\Phi_{\lambda}}-\frac{\lambda}{\theta-\Phi_{\lambda}}\frac{Z(a+d,\theta)}{Z(a+d,\Phi_{\lambda})}\right)\!\right)\nonumber\\&&+\mathrm{e}^{-\theta d}\int_x^a\mathrm{e}^{-\int_{x}^{z}\frac{W'(\overline{\xi}(w))}{W(\overline{\xi}(w))}\,\mathrm{d}w}\left(\frac{W'(\overline{\xi}(w))}{W(\overline{\xi}(w))}Z(\overline{\xi}(z),\theta)-Z'(\overline{\xi}(z),\theta)\right)\mathrm{d}z\nonumber\\&\,=\,&\frac{W(x+d)}{W(a+d)}\left(Z^{(0,\lambda)}_{a+d}(a+d,\theta)-W^{(0,\lambda)}_{a+d}(a+d)\left(\frac{\psi(\theta)}{\theta-\Phi_{\lambda}}-\frac{\lambda}{\theta-\Phi_{\lambda}}\frac{Z(a+d,\theta)}{Z(a+d,\Phi_{\lambda})}\right)\!\right)\nonumber\\&&-\mathrm{e}^{-\theta d}\int_x^a\frac{\mathrm{d}}{\mathrm{d}z}\left(\mathrm{e}^{-\int_{x}^{z}\frac{W'(w+d)}{W(w+d)}\,\mathrm{d}w}Z(z+d,\theta)\right)\mathrm{d}z\nonumber\\&\,=\,&\frac{W(x+d)}{W(a+d)}\left(Z(a+d,\theta)-W(a+d)\left(\frac{\psi(\theta)}{\theta-\Phi_{\lambda}}-\frac{\lambda}{\theta-\Phi_{\lambda}}\frac{Z(a+d,\theta)}{Z(a+d,\Phi_{\lambda})}\right)\!\right)\nonumber\\&&-\mathrm{e}^{-\theta d}\frac{W(x+d)}{W(a+d)}Z(a+d,\theta)+\mathrm{e}^{-\theta d}Z(x+d,\theta).\nonumber\end{eqnarray}

The proof is complete.

Proof. From Theorem 3.2, we have

\begin{eqnarray}h_{4}(x)&\,=\,&\Phi_{\lambda}-\lambda\int_{\xi(x)\wedge a}^{a}\rho(y-\xi(x);\;\lambda, a-\xi(x), \overline{\xi}(x))\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\&\,=\,&\Phi_{\lambda}-\lambda\int_{-d}^a\frac{Z^{(0,\lambda)}_{a+d}(y+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}(W'_{\lambda}(x-y)-\Phi_{\lambda}W_{\lambda}(x-y))\mathrm{d}y\nonumber\\&\,=\,&\Phi_{\lambda}-\lambda\int_{-d}^a\frac{Z(y+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}(W'_{\lambda}(x-y)-\Phi_{\lambda}W_{\lambda}(x-y))\mathrm{d}y\nonumber\\&\,=\,&\frac{\Phi_{\lambda}Z_{\lambda}(x+d,\Phi_{\lambda})-\lambda\Phi_{\lambda}\int_0^{a+d}W_{\lambda}(x+d-y)Z(y,\Phi_{\lambda})\mathrm{d}y}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\nonumber\\&&-\frac{\lambda\int_0^{a+d}Z(y,\Phi_{\lambda})(W'_{\lambda}(x-y+d)-\Phi_{\lambda}W_{\lambda}(x-y+d))\mathrm{d}y}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\nonumber\\&\,=\,&\frac{1}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\left(\Phi_{\lambda}Z_{\lambda}(x+d,\Phi_{\lambda})-\lambda\int_0^{a+d}Z(y,\Phi_{\lambda})W'_{\lambda}(x-y+d)\mathrm{d}y\right)\nonumber\\&\,=\,&\frac{(Z^{(0,\lambda)}_{a+d})'(x+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\nonumber\\&\,=\,&\frac{\mathrm{d}}{\mathrm{d}x}\ln{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})},\quad x\in(a,\infty).\nonumber\end{eqnarray}

and, hence, we have

(74) \begin{eqnarray}\mathrm{e}^{\int_a^xh_4(z)\mathrm{d}z}=\mathrm{e}^{\int_a^x\frac{\mathrm{d}}{\mathrm{d}z}\ln{Z^{(0,\lambda)}_{a+d}(z+d,\Phi_{\lambda})}\,\mathrm{d}z}=\frac{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(a+d,\Phi_{\lambda})}.\end{eqnarray}

In addition, we also have

\begin{eqnarray*}g_4(x)&\,=\,&-\lambda\int_{-\infty}^{\xi(x)\wedge a}\mathrm{e}^{\theta \left(y-\xi(x)\right)}\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\nonumber\\[2pt]&&-\int_{\xi(x)\wedge a}^{a}\hspace{-0.1cm}\frac{\lambda\left(Z(y-\xi(x),\theta)-Z(y-\xi(x),\Phi_{\lambda})\frac{\widetilde{Z}(a-\xi(x),\Phi_{\lambda}, \theta)}{\widetilde{Z}(a-\xi(x),\Phi_{\lambda}, \Phi_{\lambda})}\right)}{\lambda-\psi(\theta)}\nonumber\\[2pt]&&\times\lambda\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y+\lambda\upsilon(\overline{\xi}(x);\;\theta,a-\xi(x))\nonumber\\[2pt]&&\times\int_{\xi(x)\wedge a}^{a}\rho(y-\xi(x);\;\lambda, a-\xi(x), \overline{\xi}(x))\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\nonumber\\[2pt]&\,=\,&-\lambda\int_{-\infty}^{-d}\mathrm{e}^{\theta \left(y+d\right)}\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\nonumber\\[2pt]&&-\int_{-d}^{a}\hspace{-0.1cm}\frac{\lambda\left(Z(y+d,\theta)-Z(y+d,\Phi_{\lambda})\frac{\widetilde{Z}(a+d,\Phi_{\lambda}, \theta)}{\widetilde{Z}(a+d,\Phi_{\lambda}, \Phi_{\lambda})}\right)}{\lambda-\psi(\theta)}\nonumber\\[2pt]&&\times\lambda\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\nonumber\\[2pt]&&+\frac{\lambda^2}{\lambda-\psi(\theta)}\left(Z^{(0,\lambda)}_{a+d}(x+d,\theta)-\frac{\widetilde{Z}(a+d,\Phi_{\lambda}, \theta)}{\widetilde{Z}(a+d,\Phi_{\lambda}, \Phi_{\lambda})}Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})\right)\nonumber\\[2pt]&&\times\int_{-d}^{a}\frac{Z(y+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\nonumber\\[2pt]&\,=\,&\lambda\Phi_{\lambda}\mathrm{e}^{\theta(x+d)}\int_{x+d}^{\infty}\mathrm{e}^{-\theta y}W_{\lambda}(y)\mathrm{d}y-\lambda\mathrm{e}^{\theta(x+d)}\int_{x+d}^{\infty}\mathrm{e}^{-\theta y}W'_{\lambda}(y)\mathrm{d}y \nonumber\\[2pt]&&-\int_{-d}^{a}\hspace{-0.1cm}\frac{\lambda\left(Z(y+d,\theta)\right)}{\lambda-\psi(\theta)}\lambda\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right) \,\mathrm{d}y \nonumber\\[2pt]&&+\frac{\lambda^2}{\lambda-\psi(\theta)}Z^{(0,\lambda)}_{a+d}(x+d,\theta)\int_{-d}^{a}\frac{Z(y+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\left({-}\Phi_{\lambda}W_{\lambda}(x-y)+W_{\lambda}'(x-y)\right)\mathrm{d}y\nonumber\\[2pt]&\,=\,&\lambda\Phi_{\lambda}\mathrm{e}^{\theta d}\frac{Z_{\lambda}(x+d,\theta)}{\psi(\theta)-\lambda}-\lambda W_{\lambda}(x+d)+\lambda\theta\mathrm{e}^{\theta y}\frac{Z_{\lambda}(x+d,\theta)}{\psi(\theta)-\lambda}\nonumber\\[2pt]&&+\frac{\lambda}{\lambda-\psi(\theta)}\left(\int_0^{a+d}\lambda Z(y,\theta)\left(\Phi_{\lambda}W_{\lambda}(x+d-y)-W'_{\lambda}(x+d-y)\right)\mathrm{d}y\right)\nonumber\\[2pt]&&-\frac{\lambda}{\lambda-\psi(\theta)}\frac{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\!\left(\Phi_{\lambda}Z_{\lambda}(x+d,\Phi_{\lambda})-\lambda\int_0^{a+d}Z(y,\Phi_{\lambda})W'_{\lambda}(x+d-y)\mathrm{d}y\!\right)\nonumber\\[2pt]&&+\frac{\lambda\Phi_{\lambda}}{\lambda-\psi(\theta)}\frac{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\left(Z_{\lambda}(x+d,\Phi_{\lambda})-\lambda\int_0^{a+d}Z(y,\Phi_{\lambda})W_{\lambda}(x+d-y)\mathrm{d}y\right)\nonumber\end{eqnarray*}

(75) \begin{eqnarray}&\,=\,&\lambda \mathrm{e}^{\theta d}Z_{\lambda}(x+d,\theta)\frac{\Phi_{\lambda}-\theta}{\psi(\theta)-\lambda}+\lambda W_{\lambda}(x+d) \nonumber\\[5pt]&&+\frac{\lambda Z^{(0,\lambda)}_{a+d}(x+d,\theta)}{\lambda-\psi(\theta)}\left(\Phi_{\lambda}\frac{Z_{\lambda}(x+d,\theta)}{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}-\Phi_{\lambda}+\frac{(Z^{(0,\lambda)}_{a+d})'(x+d,\theta)}{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}-\frac{Z'_{\lambda}(x+d,\theta)}{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}\right)\nonumber\\[5pt]&&+\frac{\lambda Z^{(0,\lambda)}_{a+d}(x+d,\theta)}{\lambda-\psi(\theta)}\left(\Phi_{\lambda}-\frac{(Z^{(0,\lambda)}_{a+d})'(x+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\right)\nonumber\\[5pt]&\,=\,&\lambda Z_{\lambda}(x+d,\theta)\frac{\Phi_{\lambda}-\theta}{\psi(\theta)-\lambda}(\mathrm{e}^{\theta d}-1)\nonumber\\[5pt]&&+\frac{\lambda Z^{(0,\lambda)}_{a+d}(x+d,\theta)}{\lambda-\psi(\theta)}\left(\frac{(Z^{(0,\lambda)}_{a+d})'(x+d,\theta)}{Z^{(0,\lambda)}_{a+d}(x+d,\theta)}-\frac{(Z^{(0,\lambda)}_{a+d})'(x+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\right)\!. \end{eqnarray}

It follows from (18), (74). and (75) that, for $x\in(a,\infty)$ ,

\begin{eqnarray}f_4(x;\;\theta,a)&\,=\,&-\int_{x}^{\infty}g_{4}(z)\,\mathrm{e}^{-\int_{x}^{z}h_{4}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\[5pt]&\,=\,&-\lambda(\mathrm{e}^{\theta d}-1)\frac{\Phi_{\lambda}-\theta}{\psi(\theta)-\lambda}\int_x^{\infty}Z_{\lambda}(z+d,\theta)\frac{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(z+d,\Phi_{\lambda})}\,\mathrm{d}z\nonumber\\[5pt]&&-\frac{\lambda}{\lambda-\psi(\theta)}\int_x^{\infty}\left((Z^{(0,\lambda)}_{a+d})'(z+d,\theta)-h_4(z)Z^{(0,\lambda)}_{a+d}(z+d,\theta)\right)\mathrm{e}^{-\int_{x}^{z}h_{4}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\[5pt]&\,=\,&-\lambda(\mathrm{e}^{\theta d}-1)\frac{\Phi_{\lambda}-\theta}{\psi(\theta)-\lambda}\int_x^{\infty}Z_{\lambda}(z+d,\theta)\frac{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(z+d,\Phi_{\lambda})}\,\mathrm{d}z\nonumber\\[5pt]&&-\frac{\lambda}{\lambda-\psi(\theta)}\int_x^{\infty}\frac{\mathrm{d}}{\mathrm{d}z}\left(Z^{(0,\lambda)}_{a+d}(z+d,\theta)\mathrm{e}^{-\int_{x}^{z}h_{4}(w)\mathrm{d}w}\right)\mathrm{d}z\nonumber\\[5pt]&\,=\,&-\lambda(\mathrm{e}^{\theta d}-1)\frac{\Phi_{\lambda}-\theta}{\psi(\theta)-\lambda}\int_x^{\infty}Z_{\lambda}(z+d,\theta)\frac{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(z+d,\Phi_{\lambda})}\,\mathrm{d}z\nonumber\\[5pt]&&+\frac{\lambda}{\lambda-\psi(\theta)}\left(Z^{(0,\lambda)}_{a+d}(x+d,\theta)-Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})\frac{\widetilde{Z}(a,\Phi_{\lambda},\theta)}{\widetilde{Z}(a,\Phi_{\lambda},\Phi_{\lambda})}\right)\!.\nonumber\end{eqnarray}

Furthermore, for $x\in[0,a]$ , we have

\begin{eqnarray}f_4(x;\;\theta,z)&\,=\,&\left(1-f_{1}(x;\;0,a)\right)f_{4}(a;\;\theta,a)+f_{1}(x;\;\theta,a)\nonumber\\[5pt]&\,=\,&\frac{Z(x+d,\Phi_{\lambda})}{Z(a+d,\Phi_{\lambda})}\left({-}\lambda(\mathrm{e}^{\theta d}-1)\frac{\Phi_{\lambda}-\theta}{\psi(\theta)-\lambda}\int_a^{\infty}Z_{\lambda}(z+d,\theta)\frac{Z^{(0,\lambda)}_{a+d}(a+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(z+d,\Phi_{\lambda})}\,\mathrm{d}z\right.\nonumber\\[5pt]&&\left.+\frac{\lambda}{\lambda-\psi(\theta)}\left(Z^{(0,\lambda)}_{a+d}(a+d,\theta)-Z^{(0,\lambda)}_{a+d}(a+d,\Phi_{\lambda})\frac{\widetilde{Z}(a+d,\Phi_{\lambda},\theta)}{\widetilde{Z}(a+d,\Phi_{\lambda},\Phi_{\lambda})}\right)\right)\nonumber\\[5pt]&&+\frac{\lambda}{\lambda-\psi(\theta)}\left[Z(x+d,\theta)-Z(a+d,\theta)\frac{Z(x+d,\Phi_{\lambda})}{Z(a+d,\Phi_{\lambda})}\right].\nonumber\end{eqnarray}

Finally, from Theorem 3.2, we have

\begin{eqnarray*}g_{5}(x)\nonumber&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\int_{a\vee \xi(x)}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&-\int_{\xi(x)\wedge a}^{a}\frac{\lambda^{2}}{\Phi_{\lambda}+\theta}\frac{Z(y-\xi(x),\Phi_{\lambda})}{\widetilde{Z}(a-\xi(x),\Phi_{\lambda},\Phi_{\lambda})}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&+\lambda\int_{\xi(x)\wedge a}^{a}\rho(y-\xi(x);\;\lambda, a-\xi(x), \overline{\xi}(x))\vartheta(\overline{\xi}(x);\;\theta,a-\xi(x))\nonumber\\[5pt]&&\times\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\int_{a}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&-\int_{-d}^{a}\frac{\lambda^{2}}{\Phi_{\lambda}+\theta}\frac{Z(y+d,\Phi_{\lambda})}{\widetilde{Z}(a+d,\Phi_{\lambda},\Phi_{\lambda})}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&+\lambda\int_{-d}^{a}\rho(y+d;\;\lambda, a+d, x+d)\vartheta(x+d;\;\theta,a+d) \left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right) \,\mathrm{d}y\nonumber\\[5pt]&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\int_{a}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&-\int_{-d}^{a}\frac{\lambda^{2}}{\Phi_{\lambda}+\theta}\frac{Z(y+d,\Phi_{\lambda})}{\widetilde{Z}(a+d,\Phi_{\lambda},\Phi_{\lambda})}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&+\lambda\int_{-d}^{a}\frac{Z^{(0,\lambda)}_{a+d}(y+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&\times\left(\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}{\widetilde{Z}(a+d,\Phi_{\lambda},\Phi_{\lambda})}-\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\right)\nonumber\\[5pt]&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\int_{a}^{x}\mathrm{e}^{-\theta (y-a)}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&-\lambda^2\int_{-d}^{a}\frac{Z(y+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\left(W_{\lambda}'(x-y)-\Phi_{\lambda}W_{\lambda}(x-y)\right)\mathrm{d}y\nonumber\\[5pt]&&\hspace{0.3cm}\times\left(\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\right)\nonumber\\[5pt]&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W'_{\lambda}(y)\mathrm{d}y+\lambda\Phi_{\lambda}\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\nonumber\\[5pt]&&-\lambda^2\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\int_0^{a+d}\frac{Z(y,\Phi_{\lambda})W'_{\lambda}(x+d-y)}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\,\mathrm{d}y\nonumber\\[5pt]&&+\lambda^2\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\int_0^{a+d}\frac{\Phi_{\lambda}Z(y,\Phi_{\lambda})W_{\lambda}(x+d-y)}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\,\mathrm{d}y\nonumber \\\ &\,=\,&-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W'_{\lambda}(y)\mathrm{d}y+\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\nonumber\\[5pt]&&\times\left(\frac{\Phi_{\lambda}Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})-\lambda\int_0^{a+d}Z(y,\Phi_{\lambda})W'_{\lambda}(x+d-y)\mathrm{d}y}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}\right)\nonumber\\[5pt]&\,=\,&-\lambda W(0)\mathrm{e}^{-\theta(x-a)}-\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W'_{\lambda}(y)\mathrm{d}y\nonumber\\[5pt]&&+\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\frac{(Z^{(0,\lambda)}_{a+d})'(x+d,\Phi_{\lambda})}{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})},\nonumber\end{eqnarray*}

which, together with (20), yields that for $x\in(a,\infty)$ ,

\begin{eqnarray}f_{5}(x;\;\theta,a)&\,=\,&-\int_{x}^{\infty}g_{5}(z)\,\mathrm{e}^{-\int_{x}^{z}h_{4}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\[5pt]&\,=\,&\lambda\int_x^{\infty}\left( W(0)\mathrm{e}^{-\theta(z-a)}+\mathrm{e}^{-\theta(z-a)}\int_0^{z-a}\mathrm{e}^{\theta y}W'_{\lambda}(y)\mathrm{d}y\right.\nonumber\\[5pt]&&\left.-\mathrm{e}^{-\theta(z-a)}\int_0^{z-a}\mathrm{e}^{\theta y} W_{\lambda}(y)\mathrm{d}yh_4(z)\right)\mathrm{e}^{-\int_{x}^{z}h_{4}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\[5pt]&\,=\,&\lambda\int_x^{\infty}\left( W_{\lambda}(z-a)-\theta\mathrm{e}^{-\theta(z-a)}\int_0^{z-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y\right.\nonumber\\[5pt]&&\left.-\mathrm{e}^{-\theta(z-a)}\int_0^{z-a}\mathrm{e}^{\theta y} W_{\lambda}(y)\mathrm{d}yh_4(z)\right)\mathrm{e}^{-\int_{x}^{z}h_{4}(w)\mathrm{d}w}\,\mathrm{d}z\nonumber\\[5pt]&\,=\,&\lambda\int_x^{\infty}\frac{\mathrm{d}}{\mathrm{d}z}\left(\mathrm{e}^{-\theta(z-a)}\int_0^{z-a}\mathrm{e}^{\theta y} W_{\lambda}(y)\mathrm{d}y\mathrm{e}^{-\int_{x}^{z}h_{4}(w)\mathrm{d}w}\right)\mathrm{d}z\nonumber\\[5pt]&\,=\,&\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{Z^{(0,\lambda)}_{a+d}(x+d,\Phi_{\lambda})}{\widetilde{Z}(a+d,\Phi_{\lambda},\Phi_{\lambda})}-\lambda\mathrm{e}^{-\theta(x-a)}\int_0^{x-a}\mathrm{e}^{\theta y}W_{\lambda}(y)\mathrm{d}y.\nonumber\end{eqnarray}

Hence, from (19), we have for $x\in[0,a]$

\begin{eqnarray}f_{5}(x;\;\theta,a)=\left(1-f_{1}(x;\;0,a)\right)f_{5}(a;\;\theta,a)=\frac{\lambda}{\theta+\Phi_{\lambda}}\frac{Z(x+d,\Phi_{\lambda})}{\widetilde{Z}(a+d,\Phi_{\lambda},\Phi_{\lambda})}.\nonumber\end{eqnarray}

The proof is complete.