**2. Setting and Notation**

Let (<sup>Ω</sup>, F, F = ( F*t*)*t*≥0, P) be a filtered probability space supporting a Lévy process (Kyprianou 2006, p. 2, Definition 1.1) *X* (*X* is assumed to be F-adapted and to have independent increments relative to F). The Lévy measure (Sato 1999, p. 38, Definition 8.2) of *X* is denoted by *λ*. Next, recall from Vidmar (2015) (with supp(*ν*) denoting the support (Kallenberg 1997, p. 9) of a measure *ν* defined on the Borel *σ*-field of some topological space):

**Definition 1** (Upwards skip-free Lévy chain)**.** *X is an* upwards skip-free Lévy chain*, if it is a compound Poisson process (Sato 1999, p. 18, Definition 4.2), viz. if* <sup>E</sup>[*eizXt* ] = *et* (*eizx*−<sup>1</sup>)*λ*(*dx*) *for z* ∈ R *and t* ∈ [0, <sup>∞</sup>)*, and if for some h* > 0*,* supp(*λ*) ⊂ Z*h, whereas* supp(*λ*|B((0,∞))) = {*h*}*.*

**Remark 1.** *Of course to say that X is a compound Poisson process means simply that it is a real-valued continuous-time Markov chain, vanishing a.s. at zero, with holding times exponentially distributed of rate λ*(R) *and the law of the jumps given by λ*/*λ*(R) *(Sato 1999, p. 18, Theorem 4.3).*

In the sequel, *X* will be assumed throughout an upwards skip-free Lévy chain, with *λ*({*h*}) > 0 (*h* > 0) and characteristic exponent <sup>Ψ</sup>(*p*) = (*eipx* − 1)*λ*(*dx*) (*p* ∈ R). In general, we insist on (i) every sample path of *X* being càdlàg (i.e., right-continuous, admitting left limits) and (ii) (<sup>Ω</sup>, F, F, P) satisfying the standard assumptions (i.e., the *σ*-field F is <sup>P</sup>-complete, the filtration F is right-continuous and F0 contains all P-null sets). Nevertheless, we shall, sometimes and then only provisionally, relax assumption (ii), by transferring *X* as the coordinate process onto the canonical space D*h* := {*ω* ∈ Z[0,∞) *h* : *ω* is càdlàg} of càdlàg paths, mapping [0, ∞) → Z*h*, equipping D*h* with the *σ*-algebra and natural filtration of evaluation maps; this, however, will always be made explicit. We allow *e*1 to be exponentiallydistributed,meanone,andindependentof*X*;thendefine*ep*:= *e*1/*p*(*p*∈(0,∞)\{1}).

Furthermore, for *x* ∈ R, introduce *Tx* := inf{*t* ≥ 0 : *Xt* ≥ *<sup>x</sup>*}, the first entrance time of *X* into [*x*, <sup>∞</sup>). Please note that *Tx* is an F-stopping time (Kallenberg 1997, p. 101, Theorem 6.7). The supremum or maximum (respectively infimum or minimum) process of *X* is denoted *Xt* := sup{*Xs* : *s* ∈ [0, *t*]} (respectively *Xt* := inf{*Xs* : *s* ∈ [0, *t*]}) (*t* ≥ 0). *X* ∞ := inf{*Xs* : *s* ∈ [0, ∞)} is the overall infimum.

With regard to miscellaneous general notation we have:


The geometric law geom(*p*) with success parameter *p* ∈ (0, 1] has geom(*p*)({*k*}) = *p*(<sup>1</sup> − *p*)*<sup>k</sup>* (*k* ∈ N0), 1 − *p* is then the failure parameter. The exponential law Exp(*β*) with parameter *β* > 0 is specified by the density Exp(*β*)(*dt*) = *βe*<sup>−</sup>*β<sup>t</sup>* (0,∞)(*t*)*dt*. A function *f* : [0, ∞) → [0, ∞) is said to be of exponential order, if there are {*<sup>α</sup>*, *A*} ⊂ R+, such that *f*(*x*) ≤ *Aeαx* (*x* ≥ 0); *f*(+∞) := lim*x*→∞ *f*(*x*), when this limit exists. DCT (respectively MCT) stands for the dominated (respectively monotone) convergence theorem. Finally, increasing (respectively decreasing) will mean strictly increasing (respectively strictly decreasing), nondecreasing (respectively nonincreasing) being used for the weaker alternative; we will understand *a*/0 = ±∞ for *a* ∈ ±(0, <sup>∞</sup>).
