**3. Main Results**

For the process *X*, define its first down-crossing time of level 0 and up-crossing time of level *b*, respectively, by

$$
\tau\_0^- := \inf \{ t \ge 0 : X(t) < 0 \} \text{ and } \tau\_b^+ := \inf \{ t \ge 0 : X(t) > b \}.
$$

From Kyprianou (2014), the resolvent measure corresponding to *X* is absolutely continuous with respect to the Lebesgue measure and has a version of density given by

$$q \int\_0^\infty e^{-qt} \mathbb{E}\_x(f(X(t)); t < \tau\_0^- \land \tau\_b^+) \,\mathrm{d}t$$

$$= q \int\_0^b f(y) \left( \frac{\mathcal{W}^{(q)}(\mathbf{x})}{\mathcal{W}^{(q)}(b)} \mathcal{W}^{(q)}(b-y) - \mathcal{W}^{(q)}(\mathbf{x}-y) \right) \,\mathrm{d}y, \quad \mathbf{x} \in [0, b). \tag{5}$$

In preparation for showing the main results, we first present the following Lemma 1 which gives the joint Laplace transform of *ρ*+*z* and the overshoot at *ρ*+*z* of a canonical excursion *ε* with respect to the excursion measure *n*, which is a *σ*-finite measure on the space E of canonical excursions (see Section 2).

**Lemma 1.** *For any q*, *z* > 0*, we have*

$$n\left(\mathbf{e}^{-q\boldsymbol{\rho}\_z^+ + \theta(z - \varepsilon(\boldsymbol{\rho}\_z^+))}, \mathbb{P} > z\right)$$

$$= \frac{\mathcal{W}\_+^{(q)'}(z)}{\mathcal{W}^{(q)}(z)} Z^{(q)}(z, \boldsymbol{\theta}) - \theta Z^{(q)}(z, \boldsymbol{\theta}) - (q - \boldsymbol{\psi}(\boldsymbol{\theta})) \mathcal{W}^{(q)}(z). \tag{6}$$

*In particular*

$$m\left(\mathbf{e}^{-q\rho\_z^+};\overline{\varepsilon}>z\right) = \frac{\mathcal{W}\_+^{(q)'}(z)}{\mathcal{W}^{(q)}(z)}Z^{(q)}(z) - q\mathcal{W}^{(q)}(z),$$

*and*

$$m\left(\mathbf{e}^{\theta(z-\iota(\rho\_x^+))};\overline{\varepsilon}>z\right) = \frac{\mathcal{W}\_+^{\ell}(z)}{\mathcal{W}(z)}Z(z,\theta) - \theta Z(z,\theta) + \psi(\theta)\mathcal{W}(z).$$

**Proof.** Taking use of the first result in Proposition 2 of Pistorius (2007), we can prove the desired results following the arguments in Lemma 2.2 of Kyprianou and Zhou (2009).

Proposition 1 gives the joint Laplace transform of *<sup>σ</sup>*<sup>−</sup>0 and the position of the process *U* at *<sup>σ</sup>*<sup>−</sup>0 . It is similar to Theorem 1.3 of Kyprianou and Zhou (2009) where the Lévy measure is involved in the expression. The following joint Laplace transform is expressed in terms of scale functions.

**Proposition 1.** *For any q*, *θ* > 0 *and* 0 ≤ *x* ≤ *b we have*

$$\begin{split} & \mathbb{E}\_{\mathcal{X}} \left( \mathbf{e}^{-q\tau\_{0}^{-}+\theta\mathcal{U}(\sigma\_{0}^{-})} ; \sigma\_{0}^{-} < \sigma\_{b}^{+} \right) \\ &= \int\_{x}^{b} \frac{1}{(1-\gamma(\overline{\tau\_{x}}^{-1}(z)))} \exp\left(-\int\_{x}^{z} \frac{\mathcal{W}\_{+}^{(q)'}(w)}{(1-\gamma(\overline{\tau\_{x}}^{-1}(w)))\mathcal{W}^{(q)}(w)} \,\mathrm{d}w \right) \\ & \times \left( \frac{\mathcal{W}\_{+}^{(q)'}(z)}{\mathcal{W}^{(q)}(z)} Z^{(q)}(z,\theta) - \theta Z^{(q)}(z,\theta) - (q-\psi(\theta)) \mathcal{W}^{(q)}(z) \right) \,\mathrm{d}z. \tag{7} \end{split}$$

**Proof.** By Theorems 1.1 in Kyprianou and Zhou (2009) (with minor adaptation), for 0 ≤ *x* ≤ *a*, one has

$$\mathbb{E}\_{\mathbf{x}}\left(\mathbf{e}^{-q\sigma\_{d}^{+}};\sigma\_{d}^{+}<\sigma\_{0}^{-}\right) = \exp\left(-\int\_{\mathbf{x}}^{a} \frac{\mathcal{W}\_{+}^{(q)'}(w)}{(1-\gamma(\overline{\tau\_{\mathbf{x}}}^{-1}(w)))\mathcal{W}^{(q)}(w)}\,\mathrm{d}w\right). \tag{8}$$

> For 0 ≤ *x* ≤ *b*, by Equation (8) and the compensation formula in excursion theory we have

E*x* e<sup>−</sup>*q<sup>σ</sup>*<sup>−</sup><sup>0</sup> <sup>+</sup>*θU*(*σ*<sup>−</sup>0 ); *<sup>σ</sup>*<sup>−</sup>0 < *σ*+*b* <sup>=</sup>E*x* ∑*g* e<sup>−</sup>*qg* ∏*r*<*g* **<sup>1</sup>**{*<sup>ε</sup>r*≤*γx*(*x*+*<sup>L</sup>*(*r*)), *<sup>γ</sup>x*(*x*+*<sup>L</sup>*(*g*))≤*b*} × e <sup>−</sup>*qρ*+*γx*(*x*+*<sup>L</sup>*(*g*))(*<sup>ε</sup>g*)+*<sup>θ</sup> <sup>γ</sup>x*(*x*+*<sup>L</sup>*(*g*))−*εg*(*ρ*+*γx*(*x*+*<sup>L</sup>*(*g*))(*<sup>ε</sup>g*)) **<sup>1</sup>**{*<sup>ε</sup>g*>*γx*(*x*+*<sup>L</sup>*(*g*))} <sup>=</sup>E*x* ∞0 e<sup>−</sup>*q<sup>t</sup>* ∏*r*<*t* **<sup>1</sup>**{*<sup>ε</sup>r*≤*γx*(*x*+*<sup>L</sup>*(*r*)), *<sup>γ</sup>x*(*x*+*<sup>L</sup>*(*t*))≤*b*} × E <sup>e</sup><sup>−</sup>*qρ*+*γx*(*x*+*<sup>L</sup>*(*t*))+*<sup>θ</sup> <sup>γ</sup>x*(*x*+*<sup>L</sup>*(*t*))−*<sup>ε</sup>*(*ρ*+*γx*(*x*+*<sup>L</sup>*(*t*))) **<sup>1</sup>**{*ε*>*γx*(*x*+*<sup>L</sup>*(*t*))} *n*( d*ε*) <sup>d</sup>*L*(*t*) <sup>=</sup>E*x* ⎛⎝ *γ*<sup>−</sup><sup>1</sup> *x* (*b*)−*<sup>x</sup>* 0 e<sup>−</sup>*qL*−<sup>1</sup>(*<sup>t</sup>*−) ∏*<sup>r</sup>*<*L*−<sup>1</sup>(*<sup>t</sup>*−) **<sup>1</sup>**{*<sup>ε</sup>r*≤*γx*(*x*+*<sup>L</sup>*(*r*))} × *n* <sup>e</sup><sup>−</sup>*qρ*+*γx*(*x*+*<sup>t</sup>*)+*<sup>θ</sup>*(*<sup>γ</sup>x*(*x*+*<sup>t</sup>*)−*<sup>ε</sup>*(*ρ*+*γx*(*x*+*<sup>t</sup>*)))**1**{*ε*>*γx*(*x*+*<sup>t</sup>*)} d*t* = *γ* −1 *x* (*b*)−*<sup>x</sup>* 0 exp − *<sup>γ</sup>x*(*x*+*<sup>t</sup>*) *x W*(*q*) + (*w*) (1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*w*)))*W*(*q*)(*w*) d*w* ×*n* <sup>e</sup><sup>−</sup>*qρ*+*γx*(*x*+*<sup>t</sup>*)+*<sup>θ</sup>*(*<sup>γ</sup>x*(*x*+*<sup>t</sup>*)−*<sup>ε</sup>*(*ρ*+*γx*(*x*+*<sup>t</sup>*)));*<sup>ε</sup>* > *<sup>γ</sup>x*(*<sup>x</sup>* + *t*) d*t* = *bx* exp − *sx W*(*q*) + (*w*) (1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*w*)))*W*(*q*)(*w*) d*w n* e<sup>−</sup>*qρ*+*s* <sup>+</sup>*<sup>θ</sup>*(*<sup>s</sup>*−*<sup>ε</sup>*(*ρ*+*s* ));*ε* > *s* 1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*s*)) d*<sup>s</sup>*,

which together with Equation (6) yields Equation (7).

**Remark 1.** *Let γ* ≡ 0 *in Equation (7). Then U*(*t*) = *X*(*t*) *for t* ≥ 0 *and <sup>γ</sup>x*(*z*) ≡ *z for z* ≥ *x, and by Proposition 1 we have*

$$\begin{split} & \quad \mathbb{E}\_{\mathbf{x}} \big( \mathbf{e}^{-q\tau\_{0}^{+} + \theta \mathbf{X} \left( \tau\_{0}^{-} \right)}; \tau\_{0}^{-} < \tau\_{b}^{+} \big) \\ &= \int\_{\mathbf{x}}^{b} \frac{\mathcal{W}^{(q)}(\mathbf{x})}{\mathcal{W}^{(q)}(\mathbf{s})} \left( \frac{\mathcal{W}^{(q)}\_{+}(\mathbf{s})}{\mathcal{W}^{(q)}(\mathbf{s})} \mathcal{Z}^{(q)}(\mathbf{s}, \theta) - \theta \mathcal{Z}^{(q)}(\mathbf{s}, \theta) - (q - \psi(\theta)) \mathcal{W}^{(q)}(\mathbf{s}) \right) d\mathbf{s} \\ &= -\mathcal{W}^{(q)}(\mathbf{x}) \int\_{\mathbf{x}}^{b} \frac{\mathbf{d}}{\mathbf{d}\mathbf{s}} \left( \frac{\mathcal{Z}^{(q)}(\mathbf{s}, \theta)}{\mathcal{W}^{(q)}(\mathbf{s})} \right) d\mathbf{s} \\ &= \mathcal{Z}^{(q)}(\mathbf{x}, \theta) - \frac{\mathcal{W}^{(q)}(\mathbf{x})}{\mathcal{W}^{(q)}(\mathbf{b})} \mathcal{Z}^{(q)}(\mathbf{b}, \theta), \end{split}$$

*which can be found in (8.12) (with an appropriate killing rate added) in Chapter 8 of Kyprianou (2014), or Albrecher et al. (2016).*

*Let γ* ≡ *α* ∈ (0, 1) *or <sup>γ</sup>x*(*z*) = *x* + (1 − *α*)(*z* − *x*) *in Equation (7), we have for q*, *θ* > 0 *and* 0 ≤ *x* ≤ E*x* e<sup>−</sup>*q<sup>σ</sup>*<sup>−</sup><sup>0</sup> <sup>+</sup>*θU*(*σ*<sup>−</sup>0 ); *<sup>σ</sup>*<sup>−</sup>0 < *σ*+*b* = 1 1 − *α bx W*(*q*)(*x*) *<sup>W</sup>*(*q*)(*z*) 1 1−*<sup>α</sup> W*(*q*) + (*z*) *<sup>W</sup>*(*q*)(*z*) *<sup>Z</sup>*(*q*)(*<sup>z</sup>*, *θ*) − *<sup>θ</sup>Z*(*q*)(*<sup>z</sup>*, *θ*) − (*q* − *<sup>ψ</sup>*(*θ*))*W*(*q*)(*z*) d*<sup>z</sup>*.

*b*

Proposition 2 gives an expression of potential density for the process *U*.

**Proposition 2.** *The potential measure corresponding to U is absolutely continuous with respect to the Lebesgue measure with density given by*

$$\begin{split} &\int\_{0}^{\infty} \mathsf{e}^{-qt} \mathbb{P}\_{\mathbf{x}} (\mathcal{U}(t) \in \mathbf{d}u, t < \sigma\_{p}^{+} \wedge \sigma\_{0}^{-}) \, \mathrm{d}t \\ &= \mathcal{W}^{(q)}(0) \frac{1}{1 - \gamma(\overline{\gamma}\_{x}^{-1}(u))} \exp\left(-\int\_{x}^{u} \frac{\mathcal{W}\_{+}^{(q)'}(w)}{(1 - \gamma(\overline{\gamma}\_{x}^{-1}(w))) \mathcal{W}^{(q)}(w)} \, \mathrm{d}w\right) \mathbf{1}\_{\left(\mathbf{x}, \boldsymbol{b}\right)}(u) \, \mathrm{d}u \\ &+ \int\_{x}^{b} \frac{1}{1 - \gamma(\overline{\gamma}\_{x}^{-1}(y))} \exp\left(-\int\_{x}^{y} \frac{\mathcal{W}\_{+}^{(q)'}(w)}{(1 - \gamma(\overline{\gamma}\_{x}^{-1}(w))) \mathcal{W}^{(q)}(w)} \, \mathrm{d}w\right) \\ &\times \left(\mathcal{W}\_{+}^{(q)'}(y-u) - \frac{\mathcal{W}\_{+}^{(q)'}(y)}{\mathcal{W}^{(q)}(y)} \mathcal{W}^{(q)}(y-u)\right) \mathbf{1}\_{\left(0, y\right)}(u) \, \mathrm{d}y \, \mathrm{d}u, \quad \mathbf{x}, u \in \left[0, b\right], q > 0. \tag{9} \end{split}$$

**Proof.** Let *eq* be an exponentially distributed random variable independent of *X* with mean 1/*q*. For any continuous, non-negative and bounded function *f* , we have

$$\int\_0^\infty q \epsilon^{-qt} \mathbb{E}\_x(f(\mathcal{U}(t)); t < \sigma\_b^+ \wedge \sigma\_0^-) \,\mathrm{d}t$$

$$\begin{split} &= \mathbb{E}\_x\left(f(\mathcal{U}(\varepsilon\_q)) \mathbf{1}\_{\{\mathcal{U}(\varepsilon\_q) < \Box(\varepsilon\_q), \varepsilon\_q < \sigma\_b^+ \wedge \sigma\_0^-\}}\right) \\ &\quad + \mathbb{E}\_x\left(\int\_0^\infty q \epsilon^{-qt} f(\mathcal{U}(t)) \mathbf{1}\_{\{\mathcal{U}(t) = \Box(t), t < \sigma\_b^+ \wedge \sigma\_0^-\}} \,\mathrm{d}t\right). \end{split} \tag{10}$$

Note that *t*0 **<sup>1</sup>**{*X*(*s*)=*X*(*s*)} d*s* = *W*(*q*)(0) *<sup>X</sup>*(*t*), see Corollary 6 in Chapter IV of Bertoin (1996). Recalling that *U*(*t*) = *U*(*t*) is equivalent to *X*(*t*) = *X*(*t*) which implies *t* = *<sup>L</sup>*−<sup>1</sup>(*L*(*t*)), we have

E*x* ∞0 *q*e<sup>−</sup>*q<sup>t</sup> f*(*U*(*t*))**<sup>1</sup>**{*U*(*t*)=*U*(*t*), *<sup>t</sup>*<*σ*+*b* <sup>∧</sup>*σ*<sup>−</sup>0 } d*t* <sup>=</sup>E*x* ∞0 *q*e<sup>−</sup>*qL*−<sup>1</sup>(*L*(*t*)) *<sup>f</sup>*(*U*(*<sup>L</sup>*−<sup>1</sup>(*L*(*t*))))**<sup>1</sup>**{*X*(*t*)=*X*(*t*), *<sup>L</sup>*−<sup>1</sup>(*L*(*t*))<sup>&</sup>lt;*σ*+*b* <sup>∧</sup>*σ*<sup>−</sup>0 }*dt* <sup>=</sup>*<sup>W</sup>*(0)<sup>E</sup>*x* ∞0 *q*e<sup>−</sup>*qL*−<sup>1</sup>(*L*(*t*)) *<sup>f</sup>*(*U*(*<sup>L</sup>*−<sup>1</sup>(*L*(*t*))))**<sup>1</sup>**{*<sup>L</sup>*−<sup>1</sup>(*L*(*t*))<*L*−<sup>1</sup>(*γ*<sup>−</sup><sup>1</sup> *x* (*b*)−*<sup>x</sup>*)<sup>∧</sup>*σ*<sup>−</sup>0 } d*Lt* = *q<sup>W</sup>*(0) *γ* −1 *x* (*b*)−*<sup>x</sup>* 0 exp − *<sup>γ</sup>x*(*x*+*<sup>t</sup>*) *x W*(*q*) + (*w*) (1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*w*)))*W*(*q*)(*w*) d*w f*(*<sup>γ</sup>x*(*<sup>x</sup>* + *t*)) d*t* = *q<sup>W</sup>*(0) *bx* 1 (1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*y*))) exp − *yx W*(*q*) + (*w*) (1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*w*)))*W*(*q*)(*w*) d*w f*(*y*) d*y*, (11)

where Equation (8) is used in the last but one equation.

By the compensation formula in excursion theory and the memoryless property of the exponential random variable, one has

E*x f*(*U*(*eq*)); *<sup>U</sup>*(*eq*) < *<sup>U</sup>*(*eq*),*eq* < *σ*+*b* ∧ *<sup>σ</sup>*<sup>−</sup>0 <sup>=</sup>E*x* ∞0 ∑*g* e<sup>−</sup>*qg* ∏*r*<*g* **<sup>1</sup>**{*<sup>ε</sup>r*≤*γx*(*x*+*<sup>L</sup>*(*r*)), *<sup>γ</sup>x*(*x*+*<sup>L</sup>*(*g*))≤*b*} *f <sup>γ</sup>x*(*<sup>x</sup>* + *<sup>L</sup>*(*g*)) − *<sup>ε</sup>g*(*<sup>t</sup>* − *g*) ×*q*e <sup>−</sup>*q*(*<sup>t</sup>*−*g*)**1**{*g*<*t*<*g*+*ζg*∧*ρ*+*γx*(*x*+*<sup>L</sup>*(*g*))(*<sup>ε</sup>g*)}d*t* <sup>=</sup>E*x* ∞0 e<sup>−</sup>*q<sup>t</sup>* ∏*r*<*t* **<sup>1</sup>**{*<sup>ε</sup>r*≤*γx*(*x*+*<sup>L</sup>*(*r*)), *<sup>γ</sup>x*(*x*+*<sup>L</sup>*(*t*))≤*b*} × E ∞0 *q*e<sup>−</sup>*qs f* (*<sup>γ</sup>x*(*<sup>x</sup>* + *L*(*t*)) − *ε*(*s*)) **<sup>1</sup>**{*s*<sup>&</sup>lt;*ζ*<sup>∧</sup>*ρ*+*γx*(*x*+*<sup>L</sup>*(*t*))}d*<sup>s</sup> n* (d*ε*) <sup>d</sup>*L*(*t*) = *q γ* −1 *x* (*b*)−*<sup>x</sup>* 0 exp − *<sup>γ</sup>x*(*x*+*<sup>t</sup>*) *x W*(*q*) + (*w*) (1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*w*)))*W*(*q*)(*w*) d*w* × ∞0 *n* e−*qs f*(*<sup>γ</sup>x*(*<sup>x</sup>* + *t*) − *<sup>ε</sup>*(*s*))**<sup>1</sup>**{*s*<sup>&</sup>lt;*ζ*<sup>∧</sup>*ρ*+*γx*(*x*+*<sup>t</sup>*)} d*s* d*t* = *q bx* 1 1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*y*)) exp − *yx W*(*q*) + (*w*) (1 − *γ*(*γ*<sup>−</sup><sup>1</sup> *x* (*w*)))*W*(*q*)(*w*) d*w* × ∞0 *n* e<sup>−</sup>*qs f*(*y* − *<sup>ε</sup>*(*s*))**<sup>1</sup>**{*s*<sup>&</sup>lt;*ζ*<sup>∧</sup>*ρ*+*y* } d*s* d*y*. (12)

Applying the same arguments as in Equations (11) and (12), we have

$$\begin{split} & \mathbb{E}\_{\mathbf{x}} \left( f(\boldsymbol{X}(\boldsymbol{\varepsilon}\_{q})) \mathbf{1}\_{\{\boldsymbol{\varepsilon}\_{q} < \boldsymbol{\tau}\_{p}^{+} \wedge \boldsymbol{\tau}\_{0}^{-}\}} \right) \\ &= \mathbb{E}\_{\mathbf{x}} \left( f(\boldsymbol{X}(\boldsymbol{\varepsilon}\_{q})) \mathbf{1}\_{\{\boldsymbol{\mathcal{X}}(\boldsymbol{\varepsilon}\_{q}) = \boldsymbol{\widetilde{X}}(\boldsymbol{\varepsilon}\_{q}), \boldsymbol{\varepsilon}\_{q} < \boldsymbol{\tau}\_{p}^{+} \wedge \boldsymbol{\tau}\_{0}^{-}\}} \right) + \mathbb{E}\_{\mathbf{x}} \left( f(\boldsymbol{X}(\boldsymbol{\varepsilon}\_{q})) \mathbf{1}\_{\{\boldsymbol{\mathcal{X}}(\boldsymbol{\varepsilon}\_{q}) < \boldsymbol{\widetilde{X}}(\boldsymbol{\varepsilon}\_{q}), \boldsymbol{\varepsilon}\_{q} < \boldsymbol{\tau}\_{p}^{+} \wedge \boldsymbol{\tau}\_{0}^{-}\}} \right) \\ &= q \int\_{\boldsymbol{x}}^{\boldsymbol{\mathsf{H}}} \frac{\mathcal{W}^{(q)}(\mathbf{x})}{\mathcal{W}^{(q)}(\boldsymbol{y})} \left( \mathcal{W}(0) f(\boldsymbol{y}) + \int\_{0}^{\infty} n \left( \mathbf{e}^{-\mathsf{q}s} f(\boldsymbol{y} - \boldsymbol{\varepsilon}(s)) \mathbf{1}\_{\{\boldsymbol{\mathcal{X}} < \boldsymbol{\rho}\_{p}^{+} \wedge \boldsymbol{\tau}\_{0}^{-}\} \right) d\mathbf{s} \right) d\mathbf{y} . \end{split} \tag{13}$$

Equating the right hand sides of Equations (5) and (13) and then differentiating the resultant equation with respect to *b* gives

$$\begin{aligned} &\frac{W^{(q)}(\mathbf{x})}{W^{(q)}(b)} \left( \mathcal{W}(0)f(b) + \int\_0^\infty n \left( \mathbf{e}^{-q\mathbf{s}} f(b - \varepsilon(\mathbf{s})) \mathbf{1}\_{\{s < \rho\_b^+ \wedge \zeta\}} \right) \, \mathrm{d}s \right) \\ &= \frac{W^{(q)}(\mathbf{x})}{W^{(q)}(b)} \left( f(b)\mathcal{W}(0) + \int\_0^b f(y) \left( \mathcal{W}\_+^{(q)'}(b - y) - \frac{W\_+^{(q)'}(b)}{W^{(q)}(b)} \mathcal{W}^{(q)}(b - y) \right) \, \mathrm{d}y \right), \end{aligned}$$

or equivalently

$$\begin{aligned} &\int\_0^\infty n \left( \mathbf{e}^{-qs} f(\mathbf{y} - \boldsymbol{\varepsilon}(\mathbf{s})) \mathbf{1}\_{\{s < \rho\_y^+ \wedge \zeta\}} \right) \, \mathrm{d}s \\ &= \int\_0^\mathcal{Y} f(\boldsymbol{w}) \left( \mathcal{W}\_+^{(q)'} (\boldsymbol{y} - \boldsymbol{w}) - \frac{\mathcal{W}\_+^{(q)'} (\boldsymbol{y})}{\mathcal{W}^{(q)}(\boldsymbol{y})} \mathcal{W}^{(q)}(\boldsymbol{y} - \boldsymbol{w}) \right) \, \mathrm{d}w \, \mathrm{d}s \end{aligned}$$

which together with Equation (12) yields

$$\begin{split} & \mathbb{E}\_{\mathbf{x}} \left( f(\mathcal{U}(\boldsymbol{e}\_{\emptyset})); \mathcal{U}(\boldsymbol{e}\_{\emptyset}) < \overline{\mathcal{U}}(\boldsymbol{e}\_{\emptyset}), \boldsymbol{e}\_{\emptyset} < \boldsymbol{\sigma}\_{b}^{+} \wedge \boldsymbol{\sigma}\_{0}^{-} \right) \\ &= q \int\_{\mathbf{x}}^{b} \frac{1}{(1 - \gamma(\overline{\gamma\_{\mathbf{x}}^{-1}}(y)))} \exp \left( - \int\_{\mathbf{x}}^{y} \frac{W\_{+}^{(q)'}(\boldsymbol{w})}{(1 - \gamma(\overline{\gamma\_{\mathbf{x}}^{-1}}(w))) W^{(q)}(\boldsymbol{w})} \, \mathbf{d}w \right) \\ & \times \int\_{0}^{y} f(\boldsymbol{w}) \left( W\_{+}^{(q)'}(\boldsymbol{y} - \boldsymbol{w}) - \frac{W\_{+}^{(q)'}(\boldsymbol{y})}{W^{(q)}(\boldsymbol{y})} W^{(q)}(\boldsymbol{y} - \boldsymbol{w}) \right) \, \mathbf{d}w \, \mathbf{d}y, \end{split}$$

which combined with Equations (10) and (11) yields Equation (9).

**Remark 2.** *Letting γ* ≡ 0 *in Equation (9), i.e., U*(*t*) = *X*(*t*) *for t* ≥ 0 *and <sup>γ</sup>x*(*z*) ≡ *z for z* ≥ *x, by Proposition 2 we have for* 0 ≤ *x*, *u* ≤ *b*

 ∞0 *e*<sup>−</sup>*<sup>q</sup>tPx*(*X*(*t*) ∈ d*<sup>u</sup>*, *t* < *σ*+*b* ∧ *<sup>σ</sup>*<sup>−</sup>0 ) d*t* = *<sup>W</sup>*(*q*)(0)*W*(*q*)(*x*)**<sup>1</sup>**(*<sup>x</sup>*,*<sup>b</sup>*)(*u*) *<sup>W</sup>*(*q*)(*u*) d*u* + *bx <sup>W</sup>*(*q*)(*x*) *<sup>W</sup>*(*q*)(*y*) *W*(*q*) + (*y* − *u*) − *W*(*q*) + (*y*) *<sup>W</sup>*(*q*)(*y*) *<sup>W</sup>*(*q*)(*y* − *u*) **<sup>1</sup>**(0,*y*)(*u*) d*y* d*u* = *<sup>W</sup>*(*q*)(0)*W*(*q*)(*x*)**<sup>1</sup>**(*<sup>x</sup>*,*<sup>b</sup>*)(*u*) *<sup>W</sup>*(*q*)(*u*) d*u* + *<sup>W</sup>*(*q*)(*x*) *bx* dd*y W*(*q*)(*y* − *u*) *<sup>W</sup>*(*q*)(*y*) **<sup>1</sup>**(0,*y*)(*u*) d*y* d*u* =*W*(*q*)(*x*) *W*(*q*)(0)**<sup>1</sup>**(*<sup>x</sup>*,*<sup>b</sup>*)(*u*) *<sup>W</sup>*(*q*)(*u*) d*u* + *bx* dd*y W*(*q*)(*y* − *u*) *<sup>W</sup>*(*q*)(*y*) d*y* <sup>d</sup>*u***1**(0,*x*)(*u*) + *bu* dd*y W*(*q*)(*y* − *u*) *<sup>W</sup>*(*q*)(*y*) d*y* <sup>d</sup>*u***1**(*<sup>x</sup>*,*<sup>b</sup>*)(*u*) = *W*(*q*)(*b* − *u*) *W*(*q*)(*b*) *<sup>W</sup>*(*q*)(*x*)**<sup>1</sup>**(*<sup>x</sup>*,*<sup>b</sup>*)(*u*) d*u* + *W*(*q*)(*b* − *u*) *W*(*q*)(*b*) − *<sup>W</sup>*(*q*)(*x* − *u*) *<sup>W</sup>*(*q*)(*x*) *<sup>W</sup>*(*q*)(*x*)**<sup>1</sup>**(0,*x*)(*u*) d*u* = *W*(*q*)(*x*) *W*(*q*)(*b*)*<sup>W</sup>*(*q*)(*<sup>b</sup>* − *u*) − *<sup>W</sup>*(*q*)(*x* − *u*) **<sup>1</sup>**(0,*b*)(*u*) d*<sup>u</sup>*,

*which recovers Equation (5).*

> *Let γ* ≡ *α* ∈ (0, 1) *or <sup>γ</sup>x*(*z*) = *x* + (1 − *α*)(*z* − *x*) *in Equation (9), we have*

$$\begin{split} &\int\_{0}^{\infty} \mathbf{e}^{-qt} \mathbb{P}\_{\mathbf{x}} (\mathcal{U}(t) \in \operatorname{d}\mu, t < \sigma\_{b}^{+} \wedge \sigma\_{0}^{-}) \operatorname{d}t \\ &= \frac{\mathcal{W}^{(q)}(0)}{1 - a} \left( \frac{\mathcal{W}^{(q)}(\mathbf{x})}{\mathcal{W}^{(q)}(\boldsymbol{u})} \right)^{\frac{1}{1 - a}} \mathbf{1}\_{\left(\mathbf{x}, b\right)}(\boldsymbol{u}) \operatorname{d}\mu + \frac{1}{1 - a} \int\_{\mathcal{X}}^{b} \left( \frac{\mathcal{W}^{(q)}(\mathbf{x})}{\mathcal{W}^{(q)}(\boldsymbol{y})} \right)^{\frac{1}{1 - a}} \\ & \qquad \times \left( \mathcal{W}^{(q)'}\_{+}(\mathcal{Y} - \boldsymbol{u}) - \frac{\mathcal{W}^{(q)'}\_{+}(\boldsymbol{y})}{\mathcal{W}^{(q)}(\boldsymbol{y})} \mathcal{W}^{(q)}(\boldsymbol{y} - \boldsymbol{u}) \right) \mathbf{1}\_{\left(0, \boldsymbol{y}\right)}(\boldsymbol{u}) \operatorname{d}\boldsymbol{y} \operatorname{d}\mu, \quad \mathbf{x}, \boldsymbol{u} \in \left[0, b\right], q > 0. \end{split}$$

**Author Contributions:** Conceptualization, X.Z.; methodology, W.W. and X.Z.; validation, W.W. and X.Z.; investigation, W.W. and X.Z.; writing—original draft preparation, W.W.; writing—review and editing, W.W. and X.Z.; supervision, X.Z.; project administration, X.Z.; funding acquisition, W.W. and X.Z.

**Funding:** This research was partly funded by the National Natural Science Foundation of China (Nos. 11601197; 11771018) and the Program for New Century Excellent Talents in Fujian Province University.

**Acknowledgments:** Wenyuan Wang thanks Concordia University where this paper was finished during his visit.

**Conflicts of Interest:** The author declare no conflict of interest.
