*4.4. Calculating Scale Functions*

In this subsection it will be assumed for notational convenience, but without loss of generality, that *h* = 1. We define:

$$\gamma := \lambda(\mathbb{R}), \quad p := \lambda(\{1\})/\gamma, \quad q\_k := \lambda(\{-k\})/\gamma, \ k \ge 1.$$

Fix *q* ≥ 0. Then denote, provisionally, *em*,*<sup>k</sup>* := <sup>E</sup>[*e*<sup>−</sup>*qTk* {*XTk*≥−*m*}], and *ek* := *<sup>e</sup>*0,*k*, where {*<sup>m</sup>*, *k*} ⊂ N0 and note that, thanks to Theorem 6, *em*,*<sup>k</sup>* = *em*+*k em* for all {*<sup>m</sup>*, *k*} ⊂ N0. Now, *e*0 = 1. Moreover, by the strong Markov property, for each *k* ∈ N0, by conditioning on <sup>F</sup>*Tk* and then on F*J*, where *J* is the time of the first jump after *Tk* (so that, conditionally on *Tk* < <sup>∞</sup>, *J* − *Tk* ∼ Exp(*γ*)):

$$\begin{array}{rcl} \mathcal{e}\_{k+1} &=& \mathbb{E}\left[\boldsymbol{\varepsilon}^{-qT\_{k}}\mathbbm{1}\_{\{\underline{X}\_{T\_{k}}\geq 0\}}\boldsymbol{\varepsilon}^{-q(f-T\_{k})}\left(\mathbb{1}(\text{next jump after }T\_{k}\text{ up})+\\ & \quad \text{Il}(\text{next jump after }T\_{k}\text{ 1 down}, \text{ then up 2 before down more than }k-1)+\cdots+\\ \end{array}\right] \end{array}$$

(next jump after *Tk k* down & then up *k* + 1 before down more than <sup>0</sup>)*e*<sup>−</sup>*q*(*Tk*+1−*J*)

$$= -\mathfrak{e}\_k \frac{\gamma}{\gamma + q} [p + q\_1 \mathfrak{e}\_{k-1,2} + \dots + q\_k \mathfrak{e}\_{0,k+1}] = \mathfrak{e}\_k \frac{\gamma}{\gamma + q} [p + q\_1 \frac{\mathfrak{e}\_{k+1}}{\mathfrak{e}\_{k-1}} + \dots + q\_k \frac{\mathfrak{e}\_{k+1}}{\mathfrak{e}\_0}].$$

Upon division by *ekek*+1, we obtain:

$$\mathcal{W}^{(q)}(k) = \frac{\gamma}{\gamma + q} [p\mathcal{W}^{(q)}(k+1) + q\_1 \mathcal{W}^{(q)}(k-1) + \dots + q\_k \mathcal{W}^{(q)}(0)].$$

Put another way, for all *k* ∈ Z+:

$$p\mathcal{W}^{(q)}(k+1) = \left(1 + \frac{q}{\gamma}\right)\mathcal{W}^{(q)}(k) - \sum\_{l=1}^{k} q\_l \mathcal{W}^{(q)}(k-l). \tag{20}$$

Coupled with the initial condition *W*(*q*)(0) = 1/(*γp*) (from Proposition 5 and Proposition 4), this is an explicit recursion scheme by which the values of *W*(*q*) obtain (cf. (De Vylder and Goovaerts 1988, sct. 4, eq. (6) & (7)) (Dickson and Waters 1991, sct. 7, eq. (7.1) & (7.5)) (Marchal 2001, p. 255, Proposition 3.1)). We can also see the vector *W*(*q*) = (*W*(*q*)(*k*))*<sup>k</sup>*∈<sup>Z</sup> as a suitable eigenvector of the transition matrix *P* associated with the jump chain of *X*. Namely, we have for all *k* ∈ Z+: 1 + *qγ W*(*q*)(*k*) = ∑*<sup>l</sup>*∈<sup>Z</sup> *PklW*(*q*)(*l*).

Now, with regard to the function *<sup>Z</sup>*(*q*), its values can be computed directly from the values of *W*(*q*) by a straightforward summation, *<sup>Z</sup>*(*q*)(*n*) = 1 + *q* ∑*<sup>n</sup>*−<sup>1</sup> *k*=0 *W*(*q*)(*k*) (*n* ∈ N0). Alternatively, (20) yields immediately its analogue, valid for each *n* ∈ Z<sup>+</sup> (make a summation ∑*<sup>n</sup>*−<sup>1</sup> *k*=0 and multiply by *q*, using Fubini's theorem for the last sum):

$$pZ^{(q)}(n+1) - p - pq\mathbb{W}^{(q)}(0) = \left(1 + \frac{q}{\gamma}\right) \left(Z^{(q)}(n) - 1\right) - \sum\_{l=1}^{n-1} q\_l (Z^{(q)}(n-l) - 1),$$

i.e., for all *k* ∈ Z+:

$$pZ^{(q)}(k+1) + \left(1 - p - \sum\_{l=1}^{k-1} q\_l\right) = \left(1 + \frac{q}{\gamma}\right)Z^{(q)}(k) - \sum\_{l=1}^{k-1} q\_l Z^{(q)}(k-l). \tag{21}$$

Again this can be seen as an eigenvalue problem. Namely, for all *k* ∈ Z+: 1 + *qγ Z*(*q*)(*k*) = ∑*<sup>l</sup>*∈<sup>Z</sup> *PklZ*(*q*)(*l*). In summary:

**Proposition 9** (Calculation of *W*(*q*) and *<sup>Z</sup>*(*q*))**.** *Let h* = 1 *and q* ≥ 0*. Seen as vectors, W*(*q*) := (*W*(*q*)(*k*))*<sup>k</sup>*∈<sup>Z</sup> *and Z*(*q*) := (*Z*(*q*)(*k*))*<sup>k</sup>*∈<sup>Z</sup> *satisfy, entry-by-entry (P being the transition matrix associated with the jump chain of X; λq* := 1 + *q*/*λ*(R)*):*

$$(P\mathcal{W}^{(q)})|\_{\mathbb{Z}\_+} = \lambda\_q \mathcal{W}^{(q)}|\_{\mathbb{Z}\_+} \text{ and } (P\mathcal{Z}^{(q)})|\_{\mathbb{Z}\_+} = \lambda\_q \mathcal{Z}^{(q)}|\_{\mathbb{Z}\_+} \tag{22}$$

*i.e.,* (20) *and* (21) *hold true for k* ∈ Z+*. Additionally, <sup>W</sup>*(*q*)|<sup>Z</sup>− = 0 *with W*(*q*)(0) = 1/*λ*({1})*, whereas <sup>Z</sup>*(*q*)|<sup>Z</sup>−= 1*.*

An alternative form of recursions (20) and (21) is as follows: *Risks* **2018**, *6*, 102

**Corollary 1.** *We have for all n* ∈ N0*:*

$$\mathcal{W}^{(q)}(n+1) = \mathcal{W}^{(q)}(0) + \sum\_{k=1}^{n+1} \mathcal{W}^{(q)}(n+1-k) \frac{q + \lambda(-\infty, -k]}{\lambda(\{1\})}, \quad \mathcal{W}^{(q)}(0) = 1/\lambda(\{1\}), \tag{23}$$

*and for Z* 5 (*q*) := *Z*(*q*) − 1*,*

$$\widehat{Z^{(q)}}(n+1) = (n+1)\frac{q}{\lambda(\{1\})} + \sum\_{k=1}^{n} \widehat{Z^{(q)}}(n+1-k) \frac{q + \lambda(-\infty, -k]}{\lambda(\{1\})}, \quad \widehat{Z^{(q)}}(0) = 0. \tag{24}$$

**Proof.** Recursion (23) obtains from (20) as follows (cf. also (Asmussen and Albrecher 2010, (proof of) Proposition XVI.1.2)):

$$\begin{split} &pW^{(q)}(n+1) + \sum\_{k=1}^{n} q\_{k}W^{(q)}(n-k) = \boldsymbol{\nu}\_{q}W^{(q)}(n), \forall n \in \mathbb{N}\_{0} \Rightarrow \\ &pW^{(q)}(k+1) + \sum\_{m=0}^{k-1} q\_{k-m}W^{(q)}(m) = \boldsymbol{\nu}\_{q}W^{(q)}(k), \forall k \in \mathbb{N}\_{0} \Rightarrow \text{ (making a summation } \sum\_{k=0}^{n}) \\ &p\sum\_{k=0}^{n} W^{(q)}(k+1) + \sum\_{k=0}^{n} \sum\_{m=0}^{k-1} q\_{k-m}W^{(q)}(m) = \boldsymbol{\nu}\_{q}\sum\_{k=0}^{n} W^{(q)}(k), \forall n \in \mathbb{N}\_{0} \Rightarrow \text{ (Fubirm's)} \\ &pW^{(q)}(n+1) + p\sum\_{k=0}^{n} W^{(q)}(k) + \sum\_{m=0}^{n-1} W^{(q)}(m) \sum\_{k=m+1}^{n} q\_{k-m} = pW^{(q)}(0) + \boldsymbol{\nu}\_{q}\sum\_{k=0}^{n} W^{(q)}(k), \forall n \in \mathbb{N}\_{0} \Rightarrow \text{ (relabelling)} \end{split}$$

$$pW^{(q)}(n+1) + p\sum\_{k=0}^{n}W^{(q)}(k) + \sum\_{k=0}^{n-1}W^{(q)}(k)\sum\_{l=1}^{n-k}q\_l = pW^{(q)}(0) + (1+q/\gamma)\sum\_{k=0}^{n}W^{(q)}(k), \forall n \in \mathbb{N}\_{0} \implies \text{(rearnings)}$$

$$W^{(q)}(n+1) = W^{(q)}(0) + \sum\_{k=0}^{n}W^{(q)}(k)\frac{q+\gamma\sum\_{m=-k+1}^{\infty}q\_l}{p\gamma}, \forall n \in \mathbb{N}\_{0} \implies \text{(relabeling)}$$

$$W^{(q)}(n+1) = W^{(q)}(0) + \sum\_{k=1}^{n+1}W^{(q)}(n+1-k)\frac{q+\gamma\sum\_{l=k}^{\infty}q\_l}{p\gamma}, \forall n \in \mathbb{N}\_{0}$$

Then (24) follows from (23) by another summation from *n* = 0 to *n* = *w* − 1, *w* ∈ N0, 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.

• **Skip-free chain**. Let *λ* = *pδ*1 + (1 − *p*)*<sup>δ</sup>*−1. Then *W*(*k*) = 1 1−2*p* - <sup>1</sup>−*<sup>p</sup> p k*+1 − 1, unless *p* = 1/2, in which case *W*(*k*) = 2(1 + *k*), *k* ∈ N0.

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,..., *λl*+<sup>1</sup> (real, or not), in such a way that the polynomial (in *x*) *p*(*x* − *<sup>λ</sup>*1)···(*<sup>x</sup>* − *<sup>λ</sup>l*+<sup>1</sup>) coincides with the characteristic polynomial of (20) (for *q* = 0):

$$p\mathbf{x}^{l+1} - \mathbf{x}^l + q\_1\mathbf{x}^{l-1} + \dots + q\_l$$

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 {*<sup>λ</sup>*1, ... , *<sup>λ</sup>l*+<sup>1</sup>}). *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*. A particular parameter set for the zeros is depicted in Figure 1 and the following is a concrete example of this procedure.

**Figure 1.** Consider the possible zeros *λ*1, *λ*2 and *λ*3 of the characteristic polynomial of (20) (with *q* = 0), when *l* := − inf supp(*λ*) = 2 and *p* = 1/2. Straightforward computation shows they are precisely those that satisfy (o) *λ*3 = 2 − *λ*1 − *λ*2; (i) (*<sup>λ</sup>*1 − <sup>1</sup>)(*<sup>λ</sup>*2 − <sup>1</sup>)(*<sup>λ</sup>*1 + *λ*2 − 1) = 0 and (ii) *λ*1*λ*2 + (*<sup>λ</sup>*1 + *<sup>λ</sup>*2)(<sup>2</sup> − *λ*1 − *<sup>λ</sup>*2) ≥ 0 & *<sup>λ</sup>*1*λ*2(<sup>2</sup> − *λ*1 − *<sup>λ</sup>*2) < 0. In the plot one has *λ*1 as the abscissa, *λ*2 as the ordinate. The shaded area (an ellipse missing the closed inner triangle) satisfies (ii), the black lines verify (i). Then *q*1 = (*<sup>λ</sup>*1*λ*2 + (*<sup>λ</sup>*1 + *<sup>λ</sup>*2)(<sup>2</sup> − *λ*1 − *<sup>λ</sup>*2))/2 and *q*2 = (−*λ*1*λ*2(<sup>2</sup> − *λ*1 − *<sup>λ</sup>*2))/2.

• **"Reverse-engineered" chain**. Let *l* = 2, *p* = 12 and, with reference to (the caption of) Figure 1, *λ*1 = 1, *λ*2 = −12 , *λ*3 = 32 . Then this corresponds (in the sense that has been made precise above) to an upwards skip-free Lévy chain with *λ*/*λ*(R) = 12 *δ*1 + 18 *δ*−<sup>1</sup> + 38 *δ*−<sup>2</sup> and with *<sup>W</sup>*(*n*) = *A* + *<sup>B</sup>*(−12 )*n* + *C*( 32 )*<sup>n</sup>*, for all *n* ∈ Z+, for some {*<sup>A</sup>*, *B*, *C*} ⊂ R. Choosing (say) *λ*(R) = 2, we have from Proposition 4, *W*(0) = 1; and then from (20), *W*(1) = 2, *W*(2) = 154 . This renders *A* = −43 , *B* = 112, *C* = 94.

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.

• **"Geometric" chain**. Assume *p* ∈ (0, <sup>1</sup>), take an *a* ∈ (0, <sup>1</sup>), and let *ql* = (1 − *p*)(<sup>1</sup> − *a*)*al*−<sup>1</sup> for *l* ∈ N. Then (20) implies for *z*(*k*) := *<sup>W</sup>*(*k*)/*a<sup>k</sup>* that *paz*(*k* + 1) = *z*(*k*) − ∑*kl*=<sup>1</sup>(<sup>1</sup> − *p*)(<sup>1</sup> − *a*)*z*(*k* − *l*)/*<sup>a</sup>*, i.e., for *γ*(*k*) := <sup>∑</sup>*kl*=<sup>0</sup> *z*(*l*) the relation *pa*<sup>2</sup>*γ*(*<sup>k</sup>* + 1) − (*a* + *pa*<sup>2</sup>)*γ*(*k*)+(<sup>1</sup> − *p* + *pa*)*γ*(*k* − 1) = 0, a homogeneous second order linear difference equation with constant coefficients. Specialize now to *p* = *a* = 12 and take *γ* = *λ*(R) = 2. Solving the difference equation with the initial conditions that are go<sup>t</sup> from the known values of *W*(0) and *W*(1) leads to *W*(*k*) = 2( 32)*k* − 1, *k* ∈ Z+.

This example is further developed in Section 5, in the context of the modeling of the capital surplus process of an insurance company.

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.

**Remark 8.** *Let L be the infinitesimal generator (Sato 1999, p. 208, Theorem 31.5) of X. It is seen from* (22)*, that for each q* ≥ 0*,* ((*L* − *<sup>q</sup>*)*W*(*q*))|<sup>R</sup>+ = ((*L* − *<sup>q</sup>*)*Z*(*q*))|<sup>R</sup>+ = 0*.*

**Corollary 2.** *For each q* ≥ 0*, the stopped processes Y and Z, defined by Yt* := *e*<sup>−</sup>*q*(*<sup>t</sup>*∧*T*<sup>−</sup>0 )*W*(*q*) ◦ *Xt*∧*T*<sup>−</sup>0 *and Zt* := *e*<sup>−</sup>*q*(*<sup>t</sup>*∧*T*<sup>−</sup>0 )*W*(*q*) ◦ *Xt*∧*T*<sup>−</sup>0 *, t* ≥ 0*, are nonnegative* P*-martingales with respect to the natural filtration* F*<sup>X</sup>* = (F *Xs*)*s*≥<sup>0</sup> *of X.*

**Proof.** We argue for the case of the process *Y*, the justification for *Z* being similar. Let (*Hk*)*k*≥1, *H*0 := 0, be the sequence of jump times of *X* (where, possibly by discarding a <sup>P</sup>-negligible set, we may insist on all of the *Tk*, *k* ∈ N0, being finite and increasing to +∞ as *k* → ∞). Let 0 ≤ *s* < *t*, *A* ∈ F *Xs* . By the MCT it will be sufficient to establish for {*l*, *k*} ⊂ N0, *l* ≤ *k*, that:

$$\mathbb{E}[\mathbb{I}(H\_l \le s < H\_{l+1})\mathbb{I}\_A \mathbb{Y}\_t \mathbb{I}(H\_k \le t < H\_{k+1})] = \mathbb{E}[\mathbb{I}(H\_l \le s < H\_{l+1})\mathbb{I}\_A \mathbb{Y}\_t \mathbb{I}(H\_k \le t < H\_{k+1})].\tag{25}$$

On the left-hand (respectively right-hand) side of (25) we may now replace *Yt* (respectively *Ys*) by *YHk* (respectively *YHl*) and then harmlessly insist on *l* < *k*. Moreover, up to a completion, F *Xs* ⊂ *σ*((*Hm* ∧ *s*, *X*(*Hm* ∧ *<sup>s</sup>*))*m*≥<sup>0</sup>). Therefore, by a *π*/*λ*-argument, we need only verify (25) for sets *A* of the form: *A* = 3*Mm*=<sup>1</sup>{*Hm* ∧ *s* ∈ *Am*}∩{*X*(*Hm* ∧ *s*) ∈ *Bm*}, *Am*, *Bm* Borel subsets of R, 1 ≤ *m* ≤ *M*, *M* ∈ N. Due to the presence of the indicator (*Hl* ≤ *s* < *Hl*+<sup>1</sup>), we may also take, without loss of generality, *M* = *l* and hence *A* ∈ F *XHl* . Furthermore, H := *<sup>σ</sup>*(*Hl*+<sup>1</sup> − *Hl*, *Hk* − *Hl*, *Hk*+<sup>1</sup> − *Hl*) is independent of F *XHl* ∨ *σ*(*YHk* ) and then E[*YHk* |F *XHl* ∨ H] = E[*YHk* |F *XHl* ] = *YHl* , P-a.s. (as follows at once from (22) of Proposition 9), whence (25) obtains.

### **5. 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*<sup>−</sup>*x* .

To make it concrete let us take the parameters as follows. The masses of the Lévy measure on the down jumps: *λ*({−*k*})=( 12 )*k*, *k* ∈ N; mass of Lévy measure on the up jump: *λ*({1}) = 1 2 + ∑∞*<sup>n</sup>*=<sup>1</sup> *n* · ( 12 )*n* = 52 /positive "safety loading" *s* := 12/; initial capital: *x* = 10. This is a special case of the "geometric" chain from Section 4.4 with *γ* = 72 , *p* = 57 and *a* = 12 (see p. 20 for *a*). Setting, for *k* ∈ N0, *γ*(*q*)(*k*) := <sup>∑</sup>*kl*=<sup>0</sup> *W*(*q*)(*l*)2*<sup>l</sup>* produces the following difference equation: <sup>5</sup>*γ*(*q*)(*<sup>k</sup>* + 1) − (19 + <sup>4</sup>*q*)*γ*(*q*)(*k*)+(<sup>18</sup> + <sup>4</sup>*q*)*γ*(*q*)(*<sup>k</sup>* − 1) = 0, *k* ∈ N. The initial conditions are *γ*(*q*)(0) = *W*(*q*)(0) = 25 and *γ*(*q*)(1) = *γ*(*q*)(0) + 2*W*(*q*)(1) = 25 + ( 25 )<sup>2</sup>(7 + <sup>2</sup>*q*). Finishing the tedious computation with the help of *Mathematica* produces the results reported in Figure 2.

**Figure 2.** (**a**): The Laplace transform *l* := ([0, ∞) \$ *q* → <sup>E</sup>[*e*<sup>−</sup>*qT*<sup>−</sup>*x* ; *T*<sup>−</sup>*x* < ∞]) for the parameter set described in the body of the text, on the interval [0, 0.2]. The probability of ruin is <sup>P</sup>(*<sup>T</sup>*<sup>−</sup>*x* < ∞) = *l*(0) = 1 − *<sup>W</sup>*(10)/*W*(∞) = 1 − *ψ*(0+)*W*(*x*) = 1 − *sW*(*x*) .= 0.28 and the mean ruin time conditionally on ruin is <sup>E</sup>[*<sup>T</sup>*<sup>−</sup>*x* |*<sup>T</sup>*<sup>−</sup>*x* < ∞] = −*l*(0+)/*l*(0) .= 21.8 (graphically this is one over where the tangent to *l* at zero meets the abscissa); (**b**): Density of *T*<sup>−</sup>*x* on {*<sup>T</sup>*<sup>−</sup>*x* < <sup>∞</sup>}, plotted on the interval [0, <sup>20</sup>], and obtained by means of numerically inverting the Laplace transform *l* (the Lebesgue integral of this density on [0, ∞) is equal to <sup>P</sup>(*<sup>T</sup>*<sup>−</sup>*x* < ∞)).

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 Mijatovi´c 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.

**Funding:** The support of the Slovene Human Resources Development and Scholarship Fund under contract number 11010-543/2011 is acknowledged.

**Acknowledgments:** I thank Andreas Kyprianou for suggesting to me some of the investigations in this paper. I am also grateful to three anonymous Referees whose comments and suggestions have helped to improve the presentation as well as the content of this paper. Finally my thanks goes to Florin Avram for inviting me to contribute to this special issue of Risks.

**Conflicts of Interest:** The author declares no conflict of interest.
