**1. Introduction**

It was shown in Vidmar (2015) that precisely two types of Lévy processes exhibit the property of non-random overshoots: those with no positive jumps a.s., and compound Poisson processes, whose jump chain is (for some *h* > 0) a random walk on Z*h* := {*hk*: *k* ∈ <sup>Z</sup>}, skip-free to the right. The latter class was then referred to as "upwards skip-free Lévy chains". Also in the same paper it was remarked that this common property which the two classes share results in a more explicit fluctuation theory (including the Wiener-Hopf factorization) than for a general Lévy process, this being rarely the case (cf. (Kyprianou 2006, p. 172, sct. 6.5.4)).

Now, with reference to existing literature on fluctuation theory, the spectrally negative case (when there are no positive jumps, a.s.) is dealt with in detail in (Bertoin 1996, chp. VII); (Sato 1999, sct. 9.46) and especially (Kyprianou 2006, chp. 8). On the other hand, no equally exhaustive treatment of the right-continuous random walk seems to have been presented thus far, but see Brown et al. (2010); Marchal (2001); Quine (2004); (De Vylder and Goovaerts 1988, sct. 4); (Dickson and Waters 1991, sct. 7); (Doney 2007, sct. 9.3); (Spitzer 2001, *passim*).<sup>1</sup> In particular, no such exposition appears forthcoming for the continuous-time analogue of such random walks, wherein the connection and analogy to the spectrally negative class of Lévy processes becomes most transparent and direct.

In the present paper, we proceed to do just that, i.e., we develop, by analogy to the spectrally negative case, a complete fluctuation theory (including theory of scale functions) for upwards skip-free Lévy chains. Indeed, the transposition of the results from the spectrally negative to the skip-free setting is mostly straightforward. Over and above this, however, and beyond what is purely analogous to the exposition of the spectrally negative case, (i) further specifics of the reflected process (Theorem 1-1),

<sup>1</sup> However, such a treatment did eventually become available (several years after this manuscript was essentially completed, but before it was published), in the preprint Avram and Vidmar (2017).

of the excursions from the supremum (Theorem 1-3) and of the inverse of the local time at the maximum (Theorem 1-4) are identified, (ii) the class of subordinators that are the descending ladder heights processes of such upwards skip-free Lévy chains is precisely characterized (Theorem 4), and (iii) a linear recursion is presented which allows us to directly compute the families of scale functions (Equations (20), (21), Proposition 9 and Corollary 1).

Application-wise, note that the classical continuous-time Bienaymé-Galton-Watson branching process is associated with upwards skip-free Lévy chains via a suitable time change (Kyprianou 2006, sct. 1.3.4). Besides, our chains feature as a natural continuous-time approximation of the more subtle spectrally negative Lévy family, that, because of its overall tractability, has been used extensively in applied probability (in particular to model the risk process of an insurance company; see the papers Avram et al. (2007); Chiu and Yin (2005); Yang and Zhang (2001) among others). This approximation point of view is developed in Mijatovi´c et al. (2014, 2015). Finally, focusing on the insurance context, the chains may be used directly to model the aggregate capital process of an insurance company, in what is a continuous-time embedding of the discrete-time compound binomial risk model (for which see Avram and Vidmar (2017); Bao and Liu (2012); Wat et al. (2018); Xiao and Guo (2007) and the references therein). We elaborate on this latter point of view in Section 5.

The organisation of the rest of this paper is as follows. Section 2 introduces the setting and notation. Then Section 3 develops the relevant fluctuation theory, in particular details of the Wiener-Hopf factorization. Section 4 deals with the two-sided exit problem and the accompanying families of scale functions. Finally, Section 5 closes with an application to the risk process of an insurance company.
