In summary:

**Theorem 3** (Wiener-Hopf factorization for upwards skip-free Lévy chains)**.** *We have the following identities in terms of ψ and* Φ*:*

*1. For every α* ≥ 0 *and β* ∈ C<sup>→</sup>*:*

$$\mathbb{E}[\exp\{-\kappa \overline{G}^\*\_{\varepsilon\_p} - \beta \overline{X}\_{\varepsilon\_p}\}] = \frac{1 - \varepsilon^{-\Phi(p)h}}{1 - \varepsilon^{-(\beta + \Phi(p+a))h}}$$

*and*

$$\mathbb{E}\left[\exp\{-\alpha \underline{G}\_{\mathbf{c}\_p} + \beta \underline{X}\_{\mathbf{c}\_p}\}\right] = \frac{p}{p + \mathfrak{a} - \mathfrak{v}(\beta)} \frac{1 - e^{(\beta - \Phi(p + a))h}}{1 - e^{-\Phi(p)h}}$$

*(the latter whenever p* + *α* = *ψ*(*β*)*; for the unique β*0 > 0 *such that ψ*(*β*0) = *p* + *α, i.e., for β*0 = <sup>Φ</sup>(*p* + *<sup>α</sup>*)*, one has the right-hand side given by ph ψ*(*β*0)(<sup>1</sup>−*e*<sup>−</sup><sup>Φ</sup>(*p*)*<sup>h</sup>*) = *ph*Φ(*p*+*α*) 1−*e*<sup>−</sup><sup>Φ</sup>(*p*)*<sup>h</sup> ).* ˆ

*2. For some* {*k*<sup>∗</sup>, *k*} ⊂ R<sup>+</sup> *and then for every α* > 0 *and β* ∈ C<sup>→</sup>*:*

$$\kappa^\*(\alpha, \beta) = k^\*(1 - e^{-(\beta + \Phi(\alpha))h})$$

*and*

$$\pounds(\alpha,\beta) = \hat{k} \frac{\alpha - \psi(\beta)}{1 - e^{(\beta - \Phi(\alpha))h}}$$

*(the latter whenever α* = *ψ*(*β*)*; for the unique β*0 > 0 *such that ψ*(*β*0) = *α, i.e., for β*0 = <sup>Φ</sup>(*α*)*, one has the right-hand side given by* ˆ *<sup>k</sup>ψ*(*β*0)/*<sup>h</sup>* = ˆ *k <sup>h</sup>*Φ(*α*)*).*

As a consequence of Theorem 3-1, we obtain the formula for the Laplace transform of the running infimum evaluated at an independent exponentially distributed random time:

$$\mathbb{E}[e^{\beta \underline{X}\_p}] = \frac{p}{p - \psi(\beta)} \frac{1 - e^{(\beta - \Phi(p)) \hbar}}{1 - e^{-\Phi(p) \hbar}} \quad \left(\beta \in \mathbb{R}\_+ \backslash \{\Phi(p)\}\right) \tag{8}$$

(and <sup>E</sup>[*e*<sup>Φ</sup>(*p*)*Xep* ] = *p*Φ(*p*)*<sup>h</sup>* 1−*e*<sup>−</sup><sup>Φ</sup>(*p*)*<sup>h</sup>* ). In particular, if *ψ*(0+) > 0, then letting *p* ↓ 0 in (8), one obtains by the DCT:

$$\mathbb{E}[e^{\beta \mathbb{X}\_{\infty}}] = \frac{c^{\beta \mathbb{h}} - 1}{\Phi'(0+)h\psi(\beta)} \quad (\beta > 0). \tag{9}$$

We obtain next from Theorem 3-2 (recall also Remark 4-1), by letting *α* ↓ 0 therein, the Laplace exponent *φ*(*β*) := − log <sup>E</sup>[*e*<sup>−</sup>*β<sup>H</sup>* ˆ 1 (*H*<sup>ˆ</sup> 1 < ∞)] of the descending ladder heights process *H*ˆ :

$$
\phi(\beta)(e^{\theta h} - e^{\Phi(0)h}) = \psi(\beta), \quad \beta \in \mathbb{R}\_+,\tag{10}
$$

where we have set for simplicity ˆ *k* = *<sup>e</sup>*<sup>−</sup><sup>Φ</sup>(0)*h*, by insisting on a suitable choice of the local time at the minimum. This gives the following characterization of the class of Laplace exponents of the descending ladder heights processes of upwards skip-free Lévy chains (cf. (Hubalek and Kyprianou 2011, Theorem 1)):

**Theorem 4.** *Let h* ∈ (0, <sup>∞</sup>)*,* {*<sup>γ</sup>*, *q*} ⊂ R+*, and* (*φk*)*<sup>k</sup>*∈<sup>N</sup> ⊂ R+*, with q* + ∑*<sup>k</sup>*∈<sup>N</sup> *φk* ∈ (0, <sup>∞</sup>)*. Then:*

*There exists (in law) an upwards skip-free Lévy chain X with values in* Z*h and with (i) γ being the killing rate of its strict ascending ladder heights process (see Remark 4-2), and (ii) φ*(*β*) = *q* + ∑∞*<sup>k</sup>*=<sup>1</sup> *φk*(<sup>1</sup> − *<sup>e</sup>*<sup>−</sup>*βkh*)*, β* ∈ R+*, being the Laplace exponent of its descending ladder heights process.*

*if and only if the following conditions are satisfied:*


$$\gamma = (1 - 1/\mathfrak{x}) \left( \phi\_1 + \mathfrak{x} \sum\_{k \in \mathbb{N}} \phi\_k \right)$$

*on the interval x* ∈ (1, <sup>∞</sup>)*, otherwise*2*; and then defining λ*1 := *q* + ∑*<sup>k</sup>*∈<sup>N</sup> *φk, λ*−*<sup>k</sup>* := *<sup>x</sup>φk* − *φ<sup>k</sup>*+1*, k* ∈ N*; it holds:*

> *λ*−*<sup>k</sup>* ≥ 0, *k* ∈ N.

*Such an X is then unique (in law), is called the parent process, its Lévy measure is given by* ∑*<sup>k</sup>*∈<sup>N</sup> *λ*−*kδ*−*kh* + *λ*1*δh, and x* = *e*<sup>Φ</sup>(0)*h.*

**Remark 5.** *Condition Theorem 4-2 is actually quite explicit. When γ* = 0 *(equivalently, the parent process does not drift to* −<sup>∞</sup>*), it simply says that the sequence* (*φk*)*<sup>k</sup>*∈<sup>N</sup> *should be nonincreasing. In the case when the parent process X drifts to* −∞ *(equivalently, γ* > 0 *(hence q* = 0*)), we might choose x* ∈ (1, ∞) *first, then* (*φk*)*<sup>k</sup>*≥1*, and finally γ.*

**Proof.** Please note that with *φ*(*β*) =: *q* + ∑∞*<sup>k</sup>*=<sup>1</sup> *φk*(<sup>1</sup> − *<sup>e</sup>*<sup>−</sup>*βkh*), *x* := *<sup>e</sup>*<sup>Φ</sup>(0)*h*, and comparing the respective Fourier components of the left and the right hand-side, (10) is equivalent to:


Moreover, the killing rate of the strict ascending ladder heights processes expresses as *λ*(R)(1 − 1/*x*), whereas (1) and (3) alone, together imply *q* + *x* ∑*<sup>k</sup>*∈<sup>N</sup> *φk* + *φ*1 = *<sup>λ</sup>*(R).

Necessity of the conditions. Remark that the strict ascending ladder heights and the descending ladder heights processes cannot simultaneously have a strictly positive killing rate. Everything else is trivial from the above (in particular, we obtain that such an *X*, when it exists, is unique, and has the stipulated Lévy measure and Φ(0)).

Sufficiency of the conditions. The compound Poisson process *X* whose Lévy measure is given by *λ* = ∑*<sup>k</sup>*∈<sup>N</sup> *λ*−*kδ*−*kh* + *λ*1*δh* (and whose Laplace exponent we shall denote *ψ*, likewise the largest zero of *ψ* will be denoted Φ(0)) constitutes an upwards skip-free Lévy chain. Moreover, since *x* = 1, unless *q* = 0, we obtain either way that *φ*(*β*)(*eβ<sup>h</sup>* − *x*) = *ψ*(*β*) with *φ*(*β*) := *q* + ∑∞*<sup>k</sup>*=<sup>1</sup> *φk*(<sup>1</sup> − *<sup>e</sup>*<sup>−</sup>*βkh*), *β* ≥ 0. Substituting in this relation *β* := (log *<sup>x</sup>*)/*h*, we obtain at once that if *γ* > 0 (so *q* = 0), that then *X* drifts to <sup>−</sup>∞, *x* = *<sup>e</sup>*<sup>Φ</sup>(0)*h*, and hence *γ* = (1 − *e*<sup>−</sup><sup>Φ</sup>(0))*λ*(R) is the killing rate of the strict ascending ladder heights process. On the other hand, when *γ* = 0, then *x* = 1, and a direct computation reveals *ψ*(0+) = *hλ*1 − ∑*<sup>k</sup>*∈<sup>N</sup> *kh*(*φk* − *φ<sup>k</sup>*+<sup>1</sup>) = *h*(*<sup>λ</sup>*1 − ∑*<sup>k</sup>*∈<sup>N</sup> *φk*) = *hq* ≥ 0. So *X* does not drift to <sup>−</sup>∞, and Φ(0) = 0, whence (again) *x* = *<sup>e</sup>*<sup>Φ</sup>(0)*h*. Also in this case, the killing rate of the strict ascending ladder heights process is 0 = (1 − *<sup>x</sup>*)*λ*(R). Finally, and regardless of whether *γ* is strictly positive or not, compared with (10), we conclude that *φ* is indeed the Laplace exponent of the descending ladder heights process of *X*.
