*3.2. Wiener-Hopf Factorization*

**Definition 2.** *We define, for t* ≥ 0*, G*∗*t* := inf{*s* ∈ [0, *t*] : *Xs* = *Xt*}*, i.e.,* P*-a.s., G*∗*t is the last time in the interval* [0, *t*] *that X attains a new maximum. Similarly we let Gt* := sup{*s* ∈ [0, *t*] : *Xs* = *Xs*} *be,* P*-a.s., the last time on* [0, *t*] *of attaining the running infimum (t* ≥ 0*).*

While the statements of the next proposition are given for the upwards skip-free Lévy chain *X*, they in fact hold true for the Wiener-Hopf factorization of *any* compound Poisson process. Moreover, they are (essentially) known in Kyprianou (2006). Nevertheless, we begin with these general observations, in order to (a) introduce further relevant notation and (b) provide the reader with the prerequisites needed to understand the remainder of this subsection. Immediately following Proposition 3, however, we particularize to our the skip-free setting.

**Proposition 3.** *Let p* > 0*. Then:*

*1. The pairs* (*G*<sup>∗</sup>*ep* , *Xep* ) *and* (*ep* − *G*∗*ep* , *Xep* − *Xep* ) *are independent and infinitely divisible, yielding the factorisation:*

$$\frac{p}{p - i\eta - \Psi(\theta)} = \Psi\_p^+(\eta, \theta) \Psi\_p^-(\eta, \theta).$$

*where for* {*<sup>θ</sup>*, *η*} ⊂ R*,*

$$\mathbb{E}\,\Psi\_p^+(\eta,\theta) := \mathbb{E}[\exp\{i\eta \overline{G}\_{\varepsilon\_p}^\* + i\theta \overline{X}\_{\varepsilon\_p}\}] \text{ and } \Psi\_p^-(\eta,\theta) := \mathbb{E}[\exp\{i\eta \underline{G}\_{\varepsilon\_p} + i\theta \underline{X}\_{\varepsilon\_p}\}].$$

*Duality:* (*ep* − *G*∗*ep* , *Xep* − *Xep* ) *is equal in distribution to* (*Gep* , −*Xep* )*.* <sup>Ψ</sup>+*p and* <sup>Ψ</sup><sup>−</sup>*p are the Wiener-Hopf factors.*

*2. The Wiener-Hopf factors may be identified as follows:*

$$\mathbb{E}[\exp\{-a\overline{G}^\*\_{\mathfrak{e}\_p} - \beta \overline{X}\_{\mathfrak{e}\_p}\}] = \frac{\kappa^\*(p,0)}{\kappa^\*(p+\alpha,\beta)}$$

*and*

$$\mathbb{E}[\exp\{-\kappa \underline{\mathbf{G}}\_{\varepsilon\_p} + \beta \underline{\mathbf{X}}\_{\varepsilon\_p}\}] = \frac{\mathbb{K}(p, 0)}{\mathbb{K}(p + \kappa, \beta)}$$

*for* {*<sup>α</sup>*, *β*} ⊂ C<sup>→</sup>*.* *3. Here, in terms of the law of X,*

$$\kappa^\*(\alpha, \beta) := k^\* \exp \left( \int\_0^\infty \int\_{(0,\infty)} (e^{-t} - e^{-\alpha t - \beta x}) \frac{1}{t} \mathbb{P}(X\_t \in dx) dt \right).$$

*and*

$$\mathfrak{k}(\mathfrak{a},\mathfrak{k}) = \hat{k} \exp\left(\int\_0^\infty \int\_{(-\infty,0]} (e^{-t} - e^{-at+\beta x}) \frac{1}{t} \mathbb{P}(X\_t \in dx) dt\right)$$

*for α* ∈ C<sup>→</sup>*, β* ∈ C→ *and some constants* {*k*<sup>∗</sup>, ˆ*k*} ⊂ R+*.*

**Proof.** These claims are contained in the remarks regarding compound Poisson processes in (Kyprianou 2006, pp. 167–68) pursuant to the proof of Theorem 6.16 therein. Analytic continuations have been effected in part Proposition 3-3 using properties of zeros of holomorphic functions (Rudin 1970, p. 209, Theorem 10.18), the theorems of Cauchy, Morera and Fubini, and finally the finiteness/integrability properties of *q*-potential measures (Sato 1999, p. 203, Theorem 30.10(ii)).
