*4.1. The Scale Function W*

It will be convenient to consider in this subsection the times at which *X* attains a new maximum. We let *D*1, *D*2 and so on, denote the depths (possibly zero, or infinity) of the excursions below these new maxima. For *k* ∈ N, it is agreed that *Dk* = +∞ if the process *X* never reaches the level (*k* − <sup>1</sup>)*h*. Then it is clear that for *y* ∈ Z+*h* , *x* ≥ 0 (cf. (Bühlmann 1970, p. 137, para. 6.2.4(a)) (Doney 2007, sct. 9.3)):

$$\begin{aligned} \mathbb{P}(\underline{\mathbf{X}}\_{\mathcal{T}\_{\mathbf{Y}}} \ge -\mathbf{x}) &= \mathbb{P}(D\_1 \le \mathbf{x}, D\_2 \le \mathbf{x} + h, \dots, D\_{\mathbf{y}/h} \le \mathbf{x} + \mathbf{y} - h) = \\\mathbb{P}(D\_1 \le \mathbf{x}) \cdot \mathbb{P}(D\_1 \le \mathbf{x} + h) \cdot \dots \cdot \mathbb{P}(D\_1 \le \mathbf{x} + \mathbf{y} - h) &= \frac{\prod\_{r=1}^{\lfloor (\mathbf{y} + \mathbf{x})/h \rfloor} \mathbb{P}(D\_1 \le (r - 1)h)}{\prod\_{r=1}^{\lfloor \mathbf{x}/h \rfloor/h} \mathbb{P}(D\_1 \le (r - 1)h)} = \frac{W(\mathbf{x})}{W(\mathbf{x} + \mathbf{y})}, \end{aligned}$$

where we have introduced (up to a multiplicative constant) the *scale function*:

$$\mathcal{W}(\mathbf{x}) := \mathbf{1} / \prod\_{r=1}^{\lfloor \mathbf{x}/h \rfloor} \mathbb{P}(D\_1 \le (r-1)h) \quad (\mathbf{x} \ge \mathbf{0}). \tag{11}$$

(When convenient, we extend *W* by 0 on (−∞, 0).)

**Remark 6.** *If needed, we can of course express* <sup>P</sup>(*<sup>D</sup>*1 ≤ *hk*)*, k* ∈ N0*, in terms of the usual excursions away from the maximum. Thus, let D* ˜ 1 *be the depth of the first excursion away from the current maximum. By the time the process attains a new maximum (that is to say h), conditionally on this event, it will make a total of N departures away from the maximum, where (with J*1 *the first jump time of X, p* := *<sup>λ</sup>*({*h*})/*λ*(R)*, p* ˜ := <sup>P</sup>(*XJ*1 = *h*|*Th* < ∞) = *p*/P(*Th* < ∞)*) N* ∼ geom(*p*˜)*. So, denoting* ˜*θk* := P(*D*˜ 1 ≤ *hk*)*, one has* <sup>P</sup>(*<sup>D</sup>*1 ≤ *hk*) = P(*Th* < ∞) ∑∞*<sup>l</sup>*=<sup>0</sup> *p*˜(<sup>1</sup> − *p*˜)*l* ˜*θlk* = *p* <sup>1</sup>−(<sup>1</sup>−*e*<sup>Φ</sup>(0)*<sup>h</sup> <sup>p</sup>*)˜*θk, k* ∈ N0*.*

The following theorem characterizes the scale function in terms of its Laplace transform.

**Theorem 5** (The scale function)**.** *For every y* ∈ Z+*hand x* ≥ 0 *one has:*

$$\mathbb{P}(\underline{X}\_{T\_y} \ge -\mathbf{x}) = \frac{W(\mathbf{x})}{W(\mathbf{x} + \mathbf{y})} \tag{12}$$

*and W* : [0, ∞) → [0, ∞) *is (up to a multiplicative constant) the unique right-continuous and piecewise continuous function of exponential order with Laplace transform:*

$$
\hat{\mathcal{W}}(\beta) = \int\_0^\infty e^{-\beta \mathbf{x}} \mathcal{W}(\mathbf{x}) d\mathbf{x} = \frac{e^{\beta \mathbf{h}} - 1}{\beta h \psi(\beta)} \quad (\beta > \Phi(0)). \tag{13}
$$

**Proof.** (For uniqueness see e.g., (Engelberg 2005, p. 14, Theorem 10). It is clear that *W* is of exponential order, simply from the definition (11).)

Suppose first *X* tends to +<sup>∞</sup>. Then, letting *y* → ∞ in (12) above, we obtain <sup>P</sup>(−*X*∞ ≤ *x*) = *<sup>W</sup>*(*x*)/*W*(+∞). Here, since the left-hand side limit exists by the DCT, is finite and non-zero at least for all large enough *x*, so does the right-hand side, and *<sup>W</sup>*(+∞) ∈ (0, <sup>∞</sup>).

Therefore *<sup>W</sup>*(*x*) = *<sup>W</sup>*(+∞)P(−*X*∞ ≤ *x*) and hence the Laplace-Stieltjes transform of *W* is given by (9)—here we consider *W* as being extended by 0 on (−∞, <sup>0</sup>):

$$\int\_{[0,\infty)} e^{-\beta x} d\mathcal{W}(x) = \mathcal{W}(+\infty) \frac{e^{\beta \mathfrak{l}} - 1}{\Phi'(0+) h \psi(\beta)} \qquad (\beta > 0).$$

Since (integration by parts (Revuz and Yor 1999, chp. 0, Proposition 4.5)) [0,∞) *e*<sup>−</sup>*βxdW*(*x*) = *β* (0,∞) *<sup>e</sup>*<sup>−</sup>*βxW*(*x*)*dx*,

$$\int\_0^\infty e^{-\beta \mathbf{x}} \mathcal{W}(\mathbf{x}) d\mathbf{x} = \frac{\mathcal{W}(+\infty)}{\Phi'(0+)} \frac{e^{\beta h} - 1}{\beta h \psi(\beta)} \quad (\beta > 0). \tag{14}$$

Suppose now that *X* oscillates. Via Remark 3, approximate *X* by the processes *X*, > 0. In (14), fix *β*, carry over everything except for *<sup>W</sup>*(+∞) Φ(0+) , divide both sides by *<sup>W</sup>*(0), and then apply this equality to *X*. Then on the left-hand side, the quantities pertaining to *X* will converge to the ones for the process *X* as ↓ 0 by the MCT. Indeed, for *y* ∈ Z+*h* , P(*XTy* = 0) = *<sup>W</sup>*(0)/*W*(*y*) and (in the obvious notation): 1/P(*XTy* = 0) ↑ 1/P(*XTy* = 0) = *<sup>W</sup>*(*y*)/*W*(0), since *X* ↓ *X*, uniformly on bounded time sets, almost surely as ↓ 0. (It is enough to have convergence for *y* ∈ Z+*h* , as this implies convergence for all *y* ≥ 0, *W* being the right-continuous piecewise constant extension of *<sup>W</sup>*|Z+*h* .) Thus we obtain in the oscillating case, for some *α* ∈ (0, ∞) which is the limit of the right-hand side as ↓ 0:

$$\int\_0^\infty e^{-\beta x} \mathcal{W}(x) dx = a \frac{e^{\beta h} - 1}{\beta h \psi(\beta)} \quad (\beta > 0). \tag{15}$$

Finally, we are left with the case when *X* drifts to <sup>−</sup>∞. We treat this case by a change of measure (see Proposition 1 and the paragraph immediately preceding it). To this end assume, provisionally, that *X* is already the coordinate process on the canonical filtered space D*h*. Then we calculate by Proposition 2-2 (for *y* ∈ Z+*h*, *x* ≥ 0):

$$\begin{split} \mathsf{P}(\underline{X}\_{T\_{\mathcal{Y}}} \geq -\mathbf{x}) &= \mathsf{P}(T\_{\mathcal{Y}} < \infty) \mathsf{P}(\underline{X}\_{T\_{\mathcal{Y}}} \geq -\mathbf{x} | T\_{\mathcal{Y}} < \infty) = e^{-\mathsf{A}(0)y} \mathsf{P}(\underline{X^{T\_{\mathcal{Y}}}} \geq -\mathbf{x} | T\_{\mathcal{Y}} < \infty) = \\ \varepsilon^{-\mathsf{A}(0)y} \mathsf{P}^{\natural}(\underline{X^{T\_{\mathcal{Y}}}} \geq -\mathbf{x}) &= \varepsilon^{-\mathsf{A}(0)y} \mathsf{P}^{\natural}(\underline{X}\_{T(y)} \geq -\mathbf{x}) = \varepsilon^{-\mathsf{A}(0)y} \mathcal{W}^{\natural}(\mathbf{x}) / \mathcal{W}^{\natural}(\mathbf{x} + \mathbf{y}), \end{split}$$

where the third equality uses the fact that (*ω* → inf{*ω*(*s*) : *s* ∈ [0, ∞)}) : (<sup>D</sup>*h*, F) → ([−∞, <sup>∞</sup>), B([−∞, ∞)) is a measurable transformation. Here *W*- is the scale function corresponding to *X* under the measure P-, with Laplace transform:

$$\int\_0^\infty e^{-\beta x} \mathcal{W}^\sharp(x) dx = \frac{e^{\beta \mathfrak{h}} - 1}{\beta \mathfrak{h} \psi(\Phi(0) + \beta)} \quad (\beta > 0).$$

Please note that the equality P(*XTy* ≥ −*<sup>x</sup>*) = *e*<sup>−</sup><sup>Φ</sup>(0)*yW*-(*x*)/*W*-(*x* + *y*) remains true if we revert back to our original *X* (no longer assumed to be in its canonical guise). This is so because we can always go from *X* to its canonical counter-part by taking an image measure. Then the law of the process, hence the Laplace exponent and the probability P(*XTy*≥ −*<sup>x</sup>*) do not change in this transformation.

Now define *W* ˜ (*x*) := *e*<sup>Φ</sup>(0)"<sup>1</sup>+*x*/*h*#*hW*-(*x*) (*x* ≥ 0). Then *W*˜ is the right-continuous piecewise-constant extension of *W* ˜ |Z+*h* . Moreover, for all *y* ∈ Z+*h* and *x* ≥ 0, (12) obtains with *W* replaced by *W* ˜ . Plugging in *x* = 0 into (12), *W* ˜ |Z*h* and *<sup>W</sup>*|<sup>Z</sup>*h* coincide up to a multiplicative constant, hence *W* ˜ and *W* do as well. Moreover, for all *β* > <sup>Φ</sup>(0), by the MCT:

$$\begin{split} \int\_{0}^{\infty} e^{-\beta x} \tilde{W}(x) dx &= \quad \epsilon^{\Phi(0)h} \sum\_{k=0}^{\infty} \int\_{kh}^{(k+1)h} e^{-\beta x} e^{\Phi(0)kh} \mathcal{W}^{\natural}(kh) dx \\ &= \quad \epsilon^{\Phi(0)h} \sum\_{k=0}^{\infty} \frac{1}{\beta} e^{-\beta kh} (1 - e^{-\beta h}) e^{\Phi(0)kh} \mathcal{W}^{\natural}(kh) \\ &= \quad \epsilon^{\Phi(0)h} \frac{\beta - \Phi(0)}{\beta} \frac{1 - e^{-\beta h}}{1 - e^{-(\beta - \Phi(0))h}} \int\_{0}^{\infty} e^{-(\beta - \Phi(0))x} \mathcal{W}^{\natural}(x) dx \\ &= \quad \epsilon^{\Phi(0)h} \frac{\beta - \Phi(0)}{\beta} \frac{1 - e^{-\beta h}}{1 - e^{-(\beta - \Phi(0))h}} \frac{\epsilon^{(\beta - \Phi(0))h} - 1}{(\beta - \Phi(0))h\psi(\beta)} = \frac{(e^{\beta h} - 1)}{\beta h\psi(\beta)} \end{split}$$

**Remark 7.** *Henceforth the normalization of the scale function W will be understood so as to enforce the validity of* (13)*.*

**Proposition 4.** *W*(0) = 1/(*hλ*({*h*}))*, and <sup>W</sup>*(+∞) = 1/*ψ*(0+) *if* Φ(0) = 0*. If* Φ(0) > 0*, then <sup>W</sup>*(+∞)=+<sup>∞</sup>*.*

**Proof.** Integration by parts and the DCT yield *W*(0) = lim*β*<sup>→</sup>∞ *βW*<sup>ˆ</sup> (*β*). (13) and another application of the DCT then show that *W*(0) = 1/(*hλ*({*h*})). Similarly, integration by parts and the MCT give the identity *<sup>W</sup>*(+∞) = lim*β*↓0 *βW*<sup>ˆ</sup> (*β*). The conclusion *<sup>W</sup>*(+∞) = 1/*ψ*(0+) is then immediate from (13) when Φ(0) = 0. If Φ(0) > 0, then the right-hand side of (13) tends to infinity as *β* ↓ Φ(0) and thus, by the MCT, necessarily *<sup>W</sup>*(+∞)=+<sup>∞</sup>.

*4.2. The Scale Functions <sup>W</sup>*(*q*)*, q* ≥ 0

**Definition 3.** *For q* ≥ 0*, let <sup>W</sup>*(*q*)(*x*) := *<sup>e</sup>*<sup>Φ</sup>(*q*)"<sup>1</sup>+*x*/*h*#*hW*Φ(*q*)(*x*) *(x* ≥ 0*), where Wc plays the role of W but for the process* (*<sup>X</sup>*, <sup>P</sup>*c*) *(c* ≥ 0*; see Proposition 1). Please note that W*(0) = *W. When convenient we extend W*(*q*) *by* 0 *on* (−∞, <sup>0</sup>)*.*

**Theorem 6.** *For each q* ≥ 0*, W*(*q*) : [0, ∞) → [0, ∞) *is the unique right-continuous and piecewise continuous function of exponential order with Laplace transform:*

$$\widehat{\mathcal{W}^{(q)}}(\beta) = \int\_0^\infty e^{-\beta x} \mathcal{W}^{(q)}(x) dx = \frac{e^{\beta \mathfrak{k}} - 1}{\beta \mathfrak{k} (\psi(\beta) - q)} \quad (\beta > \Phi(q)). \tag{16}$$

*Moreover, for all y* ∈ Z+*hand x* ≥ 0*:*

$$\mathbb{E}[\varepsilon^{-qT\_y} \mathbb{1}\_{\{\mathbb{X}\_{T\_y} \ge -x\}}] = \frac{\mathcal{W}^{(q)}(x)}{\mathcal{W}^{(q)}(x+y)}.\tag{17}$$

**Proof.** The claim regarding the Laplace transform follows from Proposition 1, Theorem 5 and Definition 3 as it did in the case of the scale function *W* (cf. final paragraph of the proof of Theorem 5). For the second assertion, let us calculate (moving onto the canonical space D*h* as usual, using Proposition 1 and noting that *XTy* = *y* on {*Ty* < ∞}):

$$\begin{split} \mathbb{E}[\varepsilon^{-qT\_{\mathcal{Y}}}\mathbb{1}\_{\{\underline{X}\_{T\_{\mathcal{Y}}}\geq -\infty\}}] &= \mathbb{E}[\varepsilon^{\Phi(q)X\_{T\_{\mathcal{Y}}}-qT\_{\mathcal{Y}}}\mathbb{1}\_{\{\underline{X}\_{T\_{\mathcal{Y}}}\geq -\infty\}}]\varepsilon^{-\Phi(q)y} = \\ \varepsilon^{-\Phi(q)y}\mathbb{1}\_{\Phi(q)}(\underline{X}\_{T\_{\mathcal{Y}}}\geq -\infty) &= \varepsilon^{-\Phi(q)y}\frac{W\_{\Phi(q)}(\mathbf{x})}{W\_{\Phi(q)}(\mathbf{x}+y)} = \frac{W^{(q)}(\mathbf{x})}{W^{(q)}(\mathbf{x}+y)}. \end{split}$$

**Proposition 5.** *For all q* > 0*: W*(*q*)(0) = 1/(*hλ*({*h*})) *and <sup>W</sup>*(*q*)(+∞)=+<sup>∞</sup>*.*

**Proof.** As in Proposition 4, *W*(*q*)(0) = lim*β*<sup>→</sup>∞ *βW* (*q*)(*β*) = 1/(*hλ*({*h*})). Since <sup>Φ</sup>(*q*) > 0, *<sup>W</sup>*(*q*)(+∞)=+∞ also follows at once from the expression for *W* (*q*).

Moreover:

**Proposition 6.** *For q* ≥ 0*:*


**Proof.** The first claim is immediate from Proposition 4, Definition 3 and Proposition 1. To handle the second claim, let us calculate, for the Laplace transform *dW* of the measure *dW*, the quantity (using integration by parts, Theorem 5 and the fact that (since *ψ*(0+) = 0) *y<sup>λ</sup>*(*dy*) = 0):

$$\lim\_{\beta \downarrow 0} \beta \widehat{d\mathcal{W}}(\beta) = \lim\_{\beta \downarrow 0} \frac{\beta^2}{\psi(\beta)} = \frac{2}{m\_2} \in [0, +\infty).$$

For:

$$\lim\_{\beta \downarrow 0} \int (e^{\beta y} - 1) \lambda(dy) / \beta^2 = \lim\_{\beta \downarrow 0} \int \frac{e^{\beta y} - \beta y - 1}{\beta^2 y^2} y^2 \lambda(dy) = \frac{m\_2}{2}\lambda$$

by the MCT, since (*u* → *e* <sup>−</sup>*<sup>u</sup>*+*u*−1 *u*<sup>2</sup> ) is nonincreasing on (0, ∞) (the latter can be checked by comparing derivatives). The claim then follows by the Karamata Tauberian Theorem (Bingham et al. 1987, p. 37, Theorem 1.7.1 with *ρ* = 1).

*4.3. The Functions <sup>Z</sup>*(*q*)*, q* ≥ 0

**Definition 4.** *For each q* ≥ 0*, let <sup>Z</sup>*(*q*)(*x*) := 1 + *q* "*x*/*h*#*h* 0 *<sup>W</sup>*(*q*)(*z*)*dz (x* ≥ 0*). When convenient we extend these functions by* 1 *on* (−∞, <sup>0</sup>)*.*

**Definition 5.** *For x* ≥ 0*, let T*<sup>−</sup>*x* := inf{*t* ≥ 0 : *Xt* < <sup>−</sup>*<sup>x</sup>*}*.*

**Proposition 7.** *In the sense of measures on the real line, for every q* > 0*:*

$$\mathcal{P}\_{-\underline{\chi}\_q} = \frac{qh}{e^{\Phi(q)h} - 1} d\mathcal{W}^{(q)} - q\mathcal{W}^{(q)}(\cdot - h) \cdot \Delta\_{\prime}$$

*where* Δ := *h* ∑∞*<sup>k</sup>*=<sup>1</sup> *δkh is the normalized counting measure on* <sup>Z</sup>++*h* ⊂ R*,* <sup>P</sup>−*Xeq is the law of* −*Xeq under* P*, and* (*W*(*q*)(· − *h*) · Δ)(*A*) = *A <sup>W</sup>*(*q*)(*y* − *h*)Δ(*dy*) *for Borel subsets A of* R*.*

**Theorem 7.** *For each x* ≥ 0*,*

$$\mathbb{E}[\varepsilon^{-qT\_x^-} \mathbb{1}\_{\{T\_x^- < \infty\}}] = Z^{(q)}(x) - \frac{qh}{\varepsilon^{\Phi(q)h} - 1} \mathcal{W}^{(q)}(x) \tag{18}$$

*when q* > 0*, and* <sup>P</sup>(*<sup>T</sup>*<sup>−</sup>*x* < ∞) = 1 − *<sup>W</sup>*(*x*)/*W*(+∞)*. The Laplace transform of <sup>Z</sup>*(*q*)*, q* ≥ 0*, is given by:*

$$\widehat{Z^{(q)}}(\beta) = \int\_0^\infty Z^{(q)}(\mathbf{x}) e^{-\beta \mathbf{x}} d\mathbf{x} = \frac{1}{\beta} \left( 1 + \frac{q}{\Psi(\beta) - q} \right), \quad (\beta > \Phi(q)). \tag{19}$$

*Proofs of Proposition 7 and Theorem 7*. First, with regard to the Laplace transform of *<sup>Z</sup>*(*q*), we have the following derivation (using integration by parts, for every *β* > <sup>Φ</sup>(*q*)):

$$\begin{split} \int\_{0}^{\infty} Z^{(q)}(\mathbf{x}) e^{-\beta \mathbf{\hat{r}}} d\mathbf{x} &= \quad \int\_{0}^{\infty} \frac{e^{-\beta \mathbf{\hat{r}}}}{\beta} dZ^{(q)}(\mathbf{x}) = \frac{1}{\beta} \left( 1 + q \sum\_{k=1}^{\infty} e^{-\beta k \mathbf{\hat{r}}} \mathcal{W}^{(q)}((k-1)h) h \right) \\ &= \quad \frac{1}{\beta} \left( 1 + \frac{q e^{-\beta h} \beta h}{1 - e^{-\beta h}} \sum\_{k=1}^{\infty} \frac{(1 - e^{-\beta h})}{\beta} e^{-\beta (k-1) h} \mathcal{W}^{(q)}((k-1)h) \right) \\ &= \quad \frac{1}{\beta} \left( 1 + q \frac{\beta h}{e^{\beta h} - 1} \widehat{\mathcal{W}^{(q)}}(\beta) \right) = \frac{1}{\beta} \left( 1 + \frac{q}{\psi(\beta) - q} \right). \end{split}$$

*Risks* **2018**, *6*, 102

Next, to prove Proposition 7, note that it will be sufficient to check the equality of the Laplace transforms (Bhattacharya and Waymire 2007, p. 109, Theorem 8.4). By what we have just shown, (8), integration by parts, and Theorem 6, we then only need to establish, for *β* > <sup>Φ</sup>(*q*):

$$\frac{q}{\psi(\beta) - q} \frac{e^{(\beta - \Phi(q))h} - 1}{1 - e^{-\Phi(q)h}} = \frac{qh}{e^{\Phi(q)h} - 1} \frac{\beta(e^{\beta h} - 1)}{(\psi(\beta) - q)\beta h} - \frac{q}{\psi(\beta) - q}$$

,

which is clear.

> Finally, let *x* ∈ Z+*h*. For *q* > 0, evaluate the measures in Proposition 7 at [0, *<sup>x</sup>*], to obtain:

$$\begin{aligned} \mathbb{E}[\varepsilon^{-qT\_x^-} \mathbb{I}\_{\{T\_x^- < \infty\}}] &= \mathbb{P}(\varepsilon\_q \ge T\_x^-) = \mathbb{P}(\underline{\mathbb{X}}\_q < -\infty) = 1 - \mathbb{P}(\underline{\mathbb{X}}\_q \ge -\infty) \\ &= 1 + q \int\_0^\chi \mathcal{W}^{(q)}(z) dz - \frac{qh}{\epsilon^{\Phi(q)h} - 1} \mathcal{W}^{(q)}(x), \end{aligned}$$

whence the claim follows. On the other hand, when *q* = 0, the following calculation is straightforward: <sup>P</sup>(*<sup>T</sup>*<sup>−</sup>*x* < ∞) = <sup>P</sup>(*<sup>X</sup>*∞ < −*<sup>x</sup>*) = 1 − <sup>P</sup>(*<sup>X</sup>*∞ ≥ −*<sup>x</sup>*) = 1 − *<sup>W</sup>*(*x*)/*W*(+∞) (we have passed to the limit *y* → ∞ in (12) and used the DCT on the left-hand side of this equality).

**Proposition 8.** *Let q* ≥ 0*, x* ≥ 0*, y* ∈ Z+*h . Then:*

$$\mathbb{E}[\mathfrak{e}^{-qT\_x^-}\mathbb{1}\_{\{T\_x^- < T\_y\}}] = Z^{(q)}(\mathfrak{x}) - Z^{(q)}(\mathfrak{x} + \mathfrak{y})\frac{\mathcal{W}^{(q)}(\mathfrak{x})}{\mathcal{W}^{(q)}(\mathfrak{x} + \mathfrak{y})}.$$

**Proof.** Observe that {*<sup>T</sup>*<sup>−</sup>*x* = *Ty*} = ∅, P-a.s. The case when *q* = 0 is immediate and indeed contained in Theorem 5, since, P-a.s., <sup>Ω</sup>\{*<sup>T</sup>*<sup>−</sup>*x* < *Ty*} = {*<sup>T</sup>*<sup>−</sup>*x* ≥ *Ty*} = {*XTy* ≥ <sup>−</sup>*<sup>x</sup>*}. For *q* > 0 we observe that by the strong Markov property, Theorem 6 and Theorem 7:

$$\begin{split} &\mathbb{E}[\epsilon^{-q\boldsymbol{T}\_{\operatorname{x}}^{-}}\mathbb{I}\_{\{\boldsymbol{T}\_{\operatorname{x}}^{-}<\boldsymbol{T}\_{\operatorname{y}}\}} = \mathbb{E}[\epsilon^{-q\boldsymbol{T}\_{\operatorname{x}}^{-}}\mathbb{I}\_{\{\boldsymbol{T}\_{\operatorname{x}}^{-}<\boldsymbol{\infty}\}}] - \mathbb{E}[\epsilon^{-q\boldsymbol{T}\_{\operatorname{x}}^{-}}\mathbb{I}\_{\{\boldsymbol{T}\_{\operatorname{y}}<\boldsymbol{T}\_{\operatorname{x}}^{-}<\boldsymbol{\infty}\}}] \\ &= \quad Z^{(q)}(\mathbf{x}) - \frac{qh\mathsf{I}}{\epsilon^{\Phi(q)h}-1} \mathcal{W}^{(q)}(\mathbf{x}) - \mathbb{E}[\epsilon^{-q\boldsymbol{T}\_{\operatorname{y}}}\mathbb{I}\_{\{\boldsymbol{T}\_{\operatorname{y}}<\boldsymbol{T}\_{\operatorname{x}}^{-}\}}] \mathbb{E}[\epsilon^{-q\boldsymbol{T}\_{\operatorname{x}}^{-}\leqslant\boldsymbol{\rho}}\mathbb{I}\_{\{\boldsymbol{T}\_{\operatorname{x}+\boldsymbol{y}}<\boldsymbol{\infty}\}}] \\ &= \quad Z^{(q)}(\mathbf{x}) - \frac{qh\mathsf{I}}{\epsilon^{\Phi(q)h}-1} \mathcal{W}^{(q)}(\mathbf{x}) - \frac{\mathcal{W}^{(q)}(\mathbf{x})}{\mathcal{W}^{(q)}(\mathbf{x}+\boldsymbol{y})} \left(Z^{(q)}(\mathbf{x}+\boldsymbol{y}) - \frac{qh}{\epsilon^{\Phi(q)h}-1} \mathcal{W}^{(q)}(\mathbf{x}+\boldsymbol{y})\right) \\ &= \quad Z^{(q)}(\mathbf{x}) - Z^{(q)}(\mathbf{x}+\boldsymbol{y}) \frac{\mathcal{W}^{(q)}(\mathbf{x})}{\mathcal{W}^{(q)}(\mathbf{x}+\boldsymbol{y})} \end{split}$$

which completes the proof.
