Fluctuation theory for upwards skip-free L\'evy chains

A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free L\'evy chains, i.e. for right-continuous random walks embedded into continuous time as compound Poisson processes. This is done by analogy to the spectrally negative class of L\'evy processes -- several results, however, can be made more explicit/exhaustive in our compound Poisson setting. In particular, the scale functions admit a linear recursion, of constant order when the support of the jump measure is bounded, by means of which they can be calculated -- some examples are considered.


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 ∈ Z}, 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). 1 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), of the excursions from the supremum (Theorem 1-3) and of the inverse of the local time at the maximum 1 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).

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 Ψ(p) = (e ipx − 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) (Ω, F , F, P) satisfying the standard assumptions (i.e., the σ-field F is P-complete, the filtration F is right-continuous and F 0 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 ∈ : 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 exponentially distributed, mean one, and independent of X; then define e p := e 1 /p (p ∈ (0, ∞)\{1}).

Fluctuation Theory
In the following section, to fully appreciate the similarity (and eventual differences) with the spectrally negative case, the reader is invited to directly compare the exposition of this subsection with that of (Bertoin 1996, sct. VII.1) and (Kyprianou 2006, sct. 8.1).
Next, note that ψ(β) tends to +∞ as β → ∞ over the reals, due to the presence of the atom of λ at h. Upon restriction to [0, ∞), ψ is strictly convex, as follows first on (0, ∞) by using differentiation under the integral sign and noting that the second derivative is strictly positive, and then extends to [0, ∞) by continuity.
Denote then by Φ(0) the largest root of ψ| [0,∞) . Indeed, 0 is always a root, and due to strict convexity, if Φ(0) > 0, then 0 and Φ(0) are the only two roots. The two cases occur, according as to whether ψ (0+) ≥ 0 or ψ (0+) < 0, which is clear. It is less obvious, but nevertheless true, that this right derivative at 0 actually exists, indeed ψ (0+) = R xλ(dx) ∈ [−∞, ∞). This follows from the fact that (e βx − 1)/β is nonincreasing as β ↓ 0 for x ∈ R − and hence the monotone convergence applies. Continuing from this, and with a similar justification, one also gets the equality ψ (0+) = x 2 λ(dx) ∈ (0, +∞] (where we agree ψ (0+) = +∞ if ψ (0+) = −∞). In any case, ψ : [Φ(0), ∞) → [0, ∞) is continuous and increasing, it is a bijection and we let Φ With these preliminaries having been established, our first theorem identifies characteristics of the reflected process, the local time of X at the maximum (for a definition of which see e.g., (Kyprianou 2006, p. 140, Definition 6.1)), its inverse, as well as the expected length of excursions and the probability of an infinite excursion therefrom (for definitions of these terms see e.g., (Kyprianou 2006, pp. 140-47); we agree that an excursion (from the maximum) starts immediately after X leaves its running maximum and ends immediately after it returns to it; by its length we mean the amount of time between these two time points).

2.
For the reflected process Y, 0 is a holding point. The actual time spent at 0 by Y, which we shall denote L, is a local time at the maximum. Its right-continuous inverse L −1 , given by L −1 t := inf{s ≥ 0 : L s > t} (for 0 ≤ t < L ∞ ; L −1 t := ∞ otherwise), is then a (possibly killed) compound Poisson subordinator with unit positive drift.

3.
Assuming that λ((−∞, 0)) > 0 to avoid the trivial case, the expected length of an excursion away from the supremum is equal to 0)) ; whereas the probability of such an excursion being infinite is Assume again λ((−∞, 0)) > 0 to avoid the trivial case. Let N, taking values in N ∪ {+∞}, be the number of jumps the chain makes before returning to its running maximum, after it has first left it (it does so with probability 1). Then the law of L −1 is given by (for θ ∈ [0, +∞)): In particular, L −1 has a killing rate of λ((−∞, 0))p * , Lévy mass λ((−∞, 0))(1 − p * ) and its jumps have the probability law on (0, +∞) given by the length of a generic excursion from the supremum, conditional on it being finite, i.e., that of an independent N-fold sum of independent Exp(λ(R))-distributed random variables, conditional on N being finite. Moreover, one has, for k ∈ N, P(N = k) = ∑ k l=1 q l p l,k , where the coefficients (p l,k ) ∞ l,k=1 satisfy the initial conditions: the recursions: and p l,k may be interpreted as the probability of X reaching level 0 starting from level −lh for the first time on precisely the k-th jump ({l, k} ⊂ N).
Proof. Theorem 1-1 is clear, since, e.g., Y transitions away from 0 at the rate at which X makes a negative jump; and from s ∈ Z + h \{0} to 0 at the rate at which X jumps up by s or more etc. Theorem 1-2 is standard (Kyprianou 2006, p. 141, Example 6.3 & p. 149, Theorem 6.10).
We next establish Theorem 1-3. Denote, provisionally, by β the expected excursion length. Furthermore, let the discrete-time Markov chain W (on the state space N 0 ) be endowed with the initial distribution w j := q j 1−p for j ∈ N, w 0 := 0; and transition matrix P, given by P 0i = δ 0i , whereas for i ≥ 1: P ij = p, if j = i − 1; P ij = q j−i , if j > i; and P ij = 0 otherwise (W jumps down with probability p, up i steps with probability q i , i ≥ 1, until it reaches 0, where it gets stuck). Further let N be the first hitting time for W of {0}, so that a typical excursion length of X is equal in distribution to an independent sum of N (possibly infinite) Exp(λ(R))-random variables. It is Wald's identity that β = (1/λ(R))E [N]. Then (in the obvious notation, where ∞ indicates the sum is inclusive of ∞), by Fubini: where k l is the mean hitting time of {0} for W, if it starts from l ∈ N 0 , as in (Norris 1997, p. 12). From the skip-free property of the chain W it is moreover transparent that k i = αi, i ∈ N 0 , for some 0 < α ≤ ∞ (with the usual convention 0 · ∞ = 0). Moreover we know (Norris 1997, p. 17, Theorem 1.3.5) that (k i : i ∈ N 0 ) is the minimal solution to k 0 = 0 and k i = 1 + ∑ ∞ j=1 P ij k j (i ∈ N). Plugging in k i = αi, the last system of linear equations is equivalent to (provided α < ∞) 0 = 1 − pα + αζ, where ζ := ∑ ∞ j=1 jq j . Thus, if ζ < p, the minimal solution to the system is k i = i/(p − ζ), i ∈ N 0 , from which β = ζ/(λ((−∞, 0))(p − ζ)) follows at once. If ζ ≥ p, clearly we must have α = +∞, hence E[N] = +∞ and thus β = +∞.
We turn our attention now to the supremum process X. First, using the lack of memory property of the exponential law and the skip-free nature of X, we deduce from the strong Markov property applied at the time T a , that for every a, b ∈ Z + h , p > 0: P(T a+b < e p ) = P(T a < e p )P(T b < e p ). In particular, for any n ∈ N 0 : P(T nh < e p ) = P(T h < e p ) n . And since for s ∈ Z + h , {T s < e p } = {X e p ≥ s} (P-a.s.) one has (for n ∈ N 0 ): P(X e p ≥ nh) = P(X e p ≥ h) n . Therefore X e p /h ∼ geom(1 − P(X e p ≥ h)).

1.
The failure probability for the geometrically distributed X e p /h is exp{−Φ(p)h} (p > 0).

2.
X drifts to +∞, oscillates or drifts to −∞ according as to whether ψ (0+) is positive, zero, or negative. In the latter case X ∞ /h has a geometric distribution with failure probability exp{−Φ(0)h}.

3.
(T nh ) n∈N 0 is a discrete-time increasing stochastic process, vanishing at 0 and having stationary independent increments up to the explosion time, which is an independent geometric random variable; it is a killed random walk.
Remark 2. Unlike in the spectrally negative case (Bertoin 1996, p. 189), the supremum process cannot be obtained from the reflected process, since the latter does not discern a point of increase in X when the latter is at its running maximum.
It remains to consider the case of drifting to +∞ (the cases being mutually exclusive and exhaustive). Indeed, X drifts to +∞, if and only if E[T s ] is finite for each s ∈ Z + h (Bertoin 1996, p. 172, Proposition VI.17). Using again the nondecreasingness of (e −βT s − 1)/β in β ∈ [0, ∞), we deduce from (1), by the monotone convergence, that one may differentiate under the integral sign, to get Finally, Theorem 2-3 is clear. Table 1 briefly summarizes for the reader's convenience some of our main findings thus far. Table 1. Connections between the quantities ψ (0+), Φ(0), Φ (0+), the behaviour of X at large times, and the behaviour of its excursions away from the running supremum (the latter when λ((−∞, 0)) > 0).
We conclude this section by offering a way to reduce the general case of an upwards skip-free Lévy chain to one which necessarily drifts to +∞. This will prove useful in the sequel. First, there is a pathwise approximation of an oscillating X, by (what is again) an upwards skip-free Lévy chain, but drifting to infinity.
Remark 3. Suppose X oscillates. Let (possibly by enlarging the probability space to accommodate for it) N be an independent Poisson process with intensity 1 and N t := N t (t ≥ 0) so that N is a Poisson process with intensity , independent of X. Define X := X + hN . Then, as ↓ 0, X converges to X, uniformly on bounded time sets, almost surely, and is clearly an upwards skip-free Lévy chain drifting to +∞.
The reduction of the case when X drifts to −∞ is somewhat more involved and is done by a change of measure. For this purpose assume until the end of this subsection, that X is already the coordinate process on the canonical space Ω = D h , equipped with the σ-algebra F and filtration F of evaluation maps (so that P coincides with the law of X on D h and F = σ(pr s : s ∈ [0, +∞)), We make this transition in order to be able to apply the Kolmogorov extension theorem in the proposition, which follows. Note, however, that we are no longer able to assume the standard conditions on (Ω, F , F, P). Notwithstanding this, (T x ) x∈R remain F-stopping times, since by the nature of the space Proposition 1 (Exponential change of measure). Let c ≥ 0. Then, demanding: this introduces a unique measure P c on F . Under the new measure, X remains an upwards skip-free Lévy chain with Laplace exponent Proof. That P c is introduced consistently as a probability measure on F follows from the Kolmogorov extension theorem (Parthasarathy 1967, p. 143, Theorem 4.2). Indeed, M := (exp{cX t − ψ(c)t}) t≥0 is a nonnegative martingale (use independence and stationarity of increments of X and the definition of the Laplace exponent), equal identically to 1 at time 0. Furthermore, for all β ∈ C → , {t, s} ⊂ R + and Λ ∈ F t : An application of the Functional Monotone Class Theorem then shows that X is indeed a Lévy process on (Ω, F , F, P c ) and its Laplace exponent under P c is as stipulated (that X 0 = 0 P c -a.s. follows from the absolute continuity of P c with respect to P on restriction to F 0 ).
Next, from the expression for ψ c , the claim regarding λ c follows at once. Then clearly X remains an upwards skip-free Lévy chain under P c , drifting to +∞, if ψ (c+) > 0.

For every
For every x ≥ 0, the stopped process X T x = (X t∧T x ) t≥0 is identical in law under the measures P and Proof. With regard to Proposition 2-1, we have as follows. Let t ≥ 0. By the Markov property of X at time t, the process X := (X t+s − X t ) s≥0 is identical in law with X on D h and independent of F t under P. Thus, letting T y := inf{t ≥ 0 : X t ≥ y} (y ∈ R), one has for Λ ∈ F t and n ∈ N 0 , by conditioning: The second term clearly converges to P (Λ) as n → ∞. The first converges to 0, because by (2) (0) ), as n → ∞, and we have the estimate We next show Proposition 2-2. Note first that X is F-progressively measurable (in particular, measurable), hence the stopped process X T x is measurable as a mapping into D h (Karatzas and Shreve 1988, p. 5, Problem 1.16).
Furthermore, by the strong Markov property, conditionally on {T x < ∞}, F T x is independent of the future increments of X after T x , hence also of {T x < ∞} for any x > x. We deduce that the law of X T x is the same under P(·|T x < ∞) as it is under P(·|T x < ∞) for any x > x. Proposition 2-2 then follows from Proposition 2-1 by letting x tend to +∞, the algebra A being sufficient to determine equality in law by a π/λ-argument.

Wiener-Hopf Factorization
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.

1.
The pairs (G * e p , X e p ) and (e p − G * e p , X e p − X e p ) are independent and infinitely divisible, yielding the factorisation: The Wiener-Hopf factors may be identified as follows:

3.
Here, in terms of the law of X, for α ∈ C → , β ∈ C → and some constants {k * ,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)).

2.
As for the strict ascending ladder heights subordinator H * := X L * −1 (on L * −1 < ∞; +∞ otherwise), L * −1 being the right-continuous inverse of L * , and L * denoting the amount of time X has spent at a new maximum, we have, thanks to the skip-free property of X, as follows. Since P(T h < ∞) = e −Φ(0)h , X stays at a newly achieved maximum each time for an Exp(λ(R))-distributed amount of time, departing it to achieve a new maximum later on with probability e −Φ(0)h , and departing it, never to achieve a new maximum thereafter, with probability 1 − e −Φ(0)h . It follows that the Laplace exponent of H * is given by: (0))h ) (where β ∈ R + ). In other words, H * /h is a killed Poisson process of intensity λ(R)e −Φ(0)h and with killing rate λ(R)(1 − e −Φ(0)h ).
Again thanks to the skip-free nature of X, we can expand on the contents of Proposition 3, by offering further details of the Wiener-Hopf factorization. Indeed, if we let N t := X t /h and T k := T kh (t ≥ 0, k ∈ N 0 ) then clearly T := (T k ) k≥0 are the arrival times of a renewal process (with a possibly defective inter-arrival time distribution) and N := (N t ) t≥0 is the 'number of arrivals' process. One also has the relation: G * t = T N t , t ≥ 0 (P-a.s.). Thus the random variables entering the Wiener-Hopf factorization are determined in terms of the renewal process (T, N).

In summary:
Theorem 3 (Wiener-Hopf factorization for upwards skip-free Lévy chains). We have the following identities in terms of ψ and Φ:
if and only if the following conditions are satisfied: 1. γq = 0.
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)).

Theory of Scale Functions
Again the reader is invited to compare the exposition of the following section with that of (Bertoin 1996, sct. VII.2) and (Kyprianou 2006, sct. 8.2), which deal with the spectrally negative case.

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 D k = +∞ if the process X never reaches the level (k − 1)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)): where we have introduced (up to a multiplicative constant) the scale function: (When convenient, we extend W by 0 on (−∞, 0).) Remark 6. If needed, we can of course express P(D 1 ≤ hk), k ∈ N 0 , in terms of the usual excursions away from the maximum. Thus, letD 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 : The following theorem characterizes the scale function in terms of its Laplace transform. Theorem 5 (The scale function). For every y ∈ Z + h and x ≥ 0 one has: and W : [0, ∞) → [0, ∞) is (up to a multiplicative constant) the unique right-continuous and piecewise continuous function of exponential order with Laplace transform: 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 +∞. Then, letting y → ∞ in (12) above, we obtain P(−X ∞ ≤ x) = W(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 W(+∞) ∈ (0, ∞).
Therefore W(x) = W(+∞)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 (−∞, 0): Since (integration by parts (Revuz and Yor 1999, chp. 0, Proposition 4.5 Suppose now that X oscillates. Via Remark 3, approximate X by the processes X , > 0. In (14), fix β, carry over everything except for W(+∞) Φ (0+) , divide both sides by W(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(X T y = 0) = W(0)/W(y) and (in the obvious notation): 1/P(X T y = 0) ↑ 1/P(X T y = 0) = W(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 W| Z + h .) Thus we obtain in the oscillating case, for some α ∈ (0, ∞) which is the limit of the right-hand side as ↓ 0: Finally, we are left with the case when X drifts to −∞. 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): where the third equality uses the fact that (ω → inf{ω(s) : Here W is the scale function corresponding to X under the measure P , with Laplace transform: Please note that the equality P(X T y ≥ −x) = e −Φ(0)y W (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(X T y ≥ −x) do not change in this transformation. Now defineW(x) := e Φ(0) 1+x/h h W (x) (x ≥ 0). ThenW is the right-continuous piecewise-constant extension ofW| Z + h . Moreover, for all y ∈ Z + h and x ≥ 0, (12) obtains with W replaced byW. Plugging in x = 0 into (12),W| Z h and W| Z h coincide up to a multiplicative constant, henceW and W do as well. Moreover, for all β > Φ(0), by the MCT:

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

The Scale Functions W
where W c plays the role of W but for the process (X, P c ) (c ≥ 0; see Proposition 1). Please note that W (0) = W. When convenient we extend W (q) by 0 on (−∞, 0).
is the unique right-continuous and piecewise continuous function of exponential order with Laplace transform: Moreover, for all y ∈ Z + h and x ≥ 0: 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 X T y = y on {T y < ∞}): Proposition 5. For all q > 0: W (q) (0) = 1/(hλ({h})) and W (q) (+∞) = +∞.
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) yλ(dy) = 0): ) 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).

The Functions Z
. When convenient we extend these functions by 1 on (−∞, 0).
Proposition 7. In the sense of measures on the real line, for every q > 0: where ∆ := h ∑ ∞ k=1 δ kh is the normalized counting measure on Z ++ h ⊂ R, P −X eq is the law of −X e q under P, and (W (q) (· − h) · ∆)(A) = A W (q) (y − h)∆(dy) for Borel subsets A of R.
Theorem 7. For each x ≥ 0, when q > 0, and P(T − x < ∞) = 1 − W(x)/W(+∞). The Laplace transform of Z (q) , q ≥ 0, is given by: Proofs of Proposition 7 and Theorem 7. First, with regard to the Laplace transform of Z (q) , we have the following derivation (using integration by parts, for every β > Φ(q)): 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 > Φ(q): which is clear. Finally, let x ∈ Z + h . For q > 0, evaluate the measures in Proposition 7 at [0, x], to obtain: whence the claim follows. On the other hand, when q = 0, the following calculation is straightforward: (12) and used the DCT on the left-hand side of this equality).
Proof. Observe that {T − x = T y } = ∅, P-a.s. The case when q = 0 is immediate and indeed contained in Theorem 5, since, P-a.s., For q > 0 we observe that by the strong Markov property, Theorem 6 and Theorem 7: which completes the proof.

Calculating Scale Functions
In this subsection it will be assumed for notational convenience, but without loss of generality, that h = 1. We define: Fix q ≥ 0. Then denote, provisionally, e m,k := E[e −qT k 1 {X T k ≥−m} ], and e k := e 0,k , where {m, k} ⊂ N 0 and note that, thanks to Theorem 6, e m,k = e m+k e m for all {m, k} ⊂ N 0 . Now, e 0 = 1. Moreover, by the strong Markov property, for each k ∈ N 0 , by conditioning on F T k and then on F J , where J is the time of the first jump after T k (so that, conditionally on T k < ∞, J − T k ∼ Exp(γ)): e k+1 = E e −qT k 1 {X T k ≥0} e −q(J−T k ) 1(next jump after T k up) + 1(next jump after T k 1 down, then up 2 before down more than k − 1) + · · · + 1(next jump after T k k down & then up k + 1 before down more than 0) e −q(T k+1 −J) = e k γ γ + q [p + q 1 e k−1,2 + · · · + q k e 0,k+1 ] = e k γ γ + q [p + q 1 e k+1 e k−1 + · · · + q k e k+1 e 0 ].
An alternative form of recursions (20) and (21) is as follows: Corollary 1. We have for all n ∈ N 0 : and for Z (q) := Z (q) − 1, Proof. Recursion (23) obtains from (20) as follows (cf. also (Asmussen and Albrecher 2010, (proof of) Proposition XVI.1.2)): Then (24) follows from (23) by another summation from n = 0 to n = w − 1, w ∈ N 0 , say, and an interchange in the order of summation for the final sum. Now, given these explicit recursions for the calculation of the scale functions, searching for those Laplace exponents of upwards skip-free Lévy chains (equivalently, their descending ladder heights processes, cf. Theorem 4), that allow for an inversion of (16) in terms of some or another (more or less exotic) special function, appears less important. This is in contrast to the spectrally negative case, see e.g., Hubalek and Kyprianou (2011).
That said, when the scale function(s) can be expressed in terms of elementary functions, this is certainly note-worthy. In particular, whenever the support of λ is bounded from below, then (20) becomes a homogeneous linear difference equation with constant coefficients of some (finite) order, which can always be solved for explicitly in terms of elementary functions (as long as one has control over the zeros of the characteristic polynomial). The minimal example of this situation is of course when X is skip-free to the left also. For simplicity let us only consider the case q = 0. Indeed one can in general reverse-engineer the Lévy measure, so that the zeros of the characteristic polynomial of (20) (with q = 0) are known a priori, as follows. Choose l ∈ N as being − inf supp(λ); p ∈ (0, 1) as representing the probability of an up-jump; and then the numbers λ 1 , . . . , λ l+1 (real, or not), in such a way that the polynomial (in x) p(x − λ 1 ) · · · (x − λ l+1 ) coincides with the characteristic polynomial of (20) (for q = 0): px l+1 − x l + q 1 x l−1 + · · · + q l of some upwards skip-free Lévy chain, which can jump down by at most (and does jump down by) l units (this imposes some set of algebraic restrictions on the elements of {λ 1 , . . . , λ l+1 }). A priori one then has access to the zeros of the characteristic polynomial, and it remains to use the linear recursion in order to determine the first l + 1 values of W, thereby finding (via solving a set of linear equations of dimension l + 1) the sought-after particular solution of (20) (with q = 0), that is W.
An example in which the support of λ is not bounded, but one can still obtain closed form expressions in terms of elementary functions, is the following.
Beyond this "geometric" case it seems difficult to come up with other Lévy measures for X that have unbounded support and for which W could be rendered explicit in terms of elementary functions.
We close this section with the following remark and corollary (cf. (Biffis and Kyprianou 2010, eq. (12)) and (Avram et al. 2004, Remark 5), respectively, for their spectrally negative analogues): for them we no longer assume that h = 1.
Corollary 2. For each q ≥ 0, the stopped processes Y and Z, defined by Y t : , t ≥ 0, are nonnegative P-martingales with respect to the natural filtration Proof. We argue for the case of the process Y, the justification for Z being similar. Let (H k ) k≥1 , H 0 := 0, be the sequence of jump times of X (where, possibly by discarding a P-negligible set, we may insist on all of the T k , k ∈ N 0 , being finite and increasing to +∞ as k → ∞). Let 0 ≤ s < t, A ∈ F X s . By the MCT it will be sufficient to establish for {l, k} ⊂ N 0 , l ≤ k, that: On the left-hand (respectively right-hand) side of (25) we may now replace Y t (respectively Y s ) by Y H k (respectively Y H l ) and then harmlessly insist on l < k. Moreover, up to a completion, F X s ⊂ σ((H m ∧ s, X(H m ∧ s)) m≥0 ). Therefore, by a π/λ-argument, we need only verify (25)

Application to the Modeling of an Insurance Company's Risk Process
Consider an insurance company receiving a steady but temporally somewhat uncertain stream of premia-the uncertainty stemming from fluctuations in the number of insurees and/or simply from the randomness of the times at which the premia are paid in-and which, independently, incurs random claims. For simplicity assume all the collected premia are of the same size h > 0 and that the claims incurred and the initial capital x ≥ 0 are all multiples of h. A possible, if somewhat simplistic, model for the aggregate capital process of such a company, net of initial capital, is then precisely the upwards skip-free Lévy chain X of Definition 1.
Fix now the X. We retain the notation of the previous sections, and in particular of Section 4.4, assuming still that h = 1 (of course this just means that we are expressing all monetary sums in the unit of the sizes of the received premia).
As an illustration we may then consider the computation of the Laplace transform (and hence, by inversion, of the density) of the time until ruin of the insurance company, which is to say of the time T − x . To make it concrete let us take the parameters as follows. The masses of the Lévy measure on the down jumps: λ({−k}) = ( 1 2 ) k , k ∈ N; mass of Lévy measure on the up jump: λ({1}) = 1 2 + ∑ ∞ n=1 n · ( 1 2 ) n = 5 2 /positive "safety loading" s := 1 2 /; initial capital: x = 10. This is a special case of the "geometric" chain from Section 4.4 with γ = 7 2 , p = 5 7 and a = 1 2 (see p. 20 for a). Setting, for k ∈ N 0 , γ (q) (k) := ∑ k l=0 W (q) (l)2 l produces the following difference equation: 5γ (q) (k + 1) − (19 + 4q)γ (q) (k) + (18 + 4q)γ (q) (k − 1) = 0, k ∈ N. The initial conditions are γ (q) (0) = W (q) (0) = 2 5 and γ (q) (1) = γ (q) (0) + 2W (q) (1) = 2 5 + ( 2 5 ) 2 (7 + 2q). Finishing the tedious computation with the help of Mathematica produces the results reported in Figure 2. On a final note, we should point out that the assumptions made above concerning the risk process are, strictly speaking, unrealistic. Indeed (i) the collected premia will typically not all be of the same size, and, moreover, (ii) the initial capital, and incurred claims will not be a multiple thereof. Besides, there is no reason to believe (iii) that the times that elapse between the accrual of premia are (approximately) i.id. exponentially distributed. Nevertheless, these objections can be reasonably addressed to some extent. For (ii) one just need to choose h small enough so that the error committed in "rounding off" the initial capital and the claims is negligible (of course even a priori the monetary units are not infinitely divisible, but e.g., h = 0.01 e, may not be the most computationally efficient unit to consider in this context). Concerning (i) and (iii) we would typically prefer to see a premium drift (with slight stochastic deviations). This can be achieved by taking λ({h}) sufficiently large: we will then be witnessing the arrival of premia with very high-intensity, which by the law of large numbers on a large enough time scale will look essentially like premium drift (but slightly stochastic), interdispersed with the arrivals of claims. This is basically an approximation of the Cramér-Lundberg model in the spirit of Mijatović et al. (2015), which however (because we are not ultimately effecting the limits h ↓ 0, λ({h}) → ∞) retains some stochasticity in the premia. Keeping this in mind, it would be interesting to see how the upwards skip-free model behaves when fitted against real data, but this investigation lies beyond the intended scope of the present text.