*3.1. Second Family of Scale Functions*

The so-called second scale functions are defined by: for each *q*, *θ* ≥ 0 and for *x* ∈ R, let

$$Z\_q(\mathbf{x}, \theta) = \mathbf{e}^{\theta \mathbf{x}} \left( 1 - (\psi(\theta) - q) \int\_0^\mathbf{x} \mathbf{e}^{-\theta y} \mathcal{W}^{(q)}(y) \mathbf{d}y \right). \tag{4}$$

Please note that for *x* ≤ 0 or for *θ* = <sup>Φ</sup>(*q*), we have *Zq*(*<sup>x</sup>*, *θ*) = <sup>e</sup>*θ<sup>x</sup>*. The second scale functions have appeared in the literature in various forms; see e.g., (Albrecher et al. 2016; Avram et al. 2015; Ivanovs and Palmowski 2012).

In what follows, *<sup>Z</sup>q*(*<sup>x</sup>*, *θ*) will represent the derivative with respect to the first argument. Consequently, for *x* > 0, we have *<sup>Z</sup>q*(*<sup>x</sup>*, *θ*) = *<sup>θ</sup>Zq*(*<sup>x</sup>*, *θ*) − (*ψ*(*θ*) − *q*)*W*(*q*)(*x*) and, for *x* < 0, we have *<sup>Z</sup>q*(*<sup>x</sup>*, *θ*) = *θ*e*θ<sup>x</sup>*.

In this paper, we will encounter the function *Zq* when *θ* = <sup>Φ</sup>(*p* + *q*), that is the function

$$Z\_q(x, \Phi(p+q)) = \mathbf{e}^{\Phi(p+q)x} \left(1 - p \int\_0^x \mathbf{e}^{-\Phi(p+q)y} W^{(q)}(y) d\mathbf{y}\right),$$

from which we deduce that, for *x* > 0,

$$Z\_q'(\mathbf{x}, \Phi(p+q)) = \Phi(p+q)Z\_q(\mathbf{x}, \Phi(p+q)) - p\mathcal{W}^{(q)}(\mathbf{x}).\tag{5}$$

Consequently, set *<sup>Z</sup>q*(0, <sup>Φ</sup>(*p* + *q*)) = <sup>Φ</sup>(*p* + *q*) − *<sup>p</sup>W*(*q*)(0). Since we assume that *p* > 0, we have that <sup>Φ</sup>(*p* + *q*) > <sup>Φ</sup>(*q*) and we can write

$$Z\_q(\mathbf{x}, \Phi(p+q)) = p \int\_0^\infty \mathbf{e}^{-\Phi(p+q)y} \mathcal{W}^{(q)}(\mathbf{x}+y) d\mathbf{y}, \quad \mathbf{x} \in \mathbb{R}.\tag{6}$$

Then, for *x* > 0, we have

$$Z\_q'(\mathbf{x}, \Phi(p+q)) = p \int\_0^\infty \mathbf{e}^{-\Phi(p+q)y} \mathcal{W}^{(q)\prime}(\mathbf{x}+y) \mathbf{d}y,\tag{7}$$

which is well defined since *W*(*q*) is differentiable almost everywhere (see e.g., Lemma 2.3 in (Kuznetsov et al. 2012)). Clearly, *x* → *Zq*(*<sup>x</sup>*, <sup>Φ</sup>(*p* + *q*)) is a non-decreasing continuous function. In fact, it will

be proved in Appendix B that if the tail of the Lévy measure is log-convex, then *Z q*(·, <sup>Φ</sup>(*p* + *q*)) is a log-convex function on (0, <sup>∞</sup>).

### *3.2. Value Function of a Barrier Strategy*

Here is the value of an arbitrary admissible barrier strategy:

**Proposition 1.** *For q*, *b* ≥ 0*, the value function associated with πb is given by*

$$v\_b(\mathbf{x}) = \begin{cases} \frac{Z\_q(\mathbf{x}, \Phi(p+q))}{Z\_q'(b, \Phi(p+q))} & \text{for } \mathbf{x} \in ( -\infty, b], \\\mathbf{x} - b + v\_b(b) & \text{for } \mathbf{x} \in (b, \infty). \end{cases} \tag{8}$$

**Proof.** This result has appeared before in the literature. See for example Equation (15) in (Albrecher and Ivanovs 2014) or Equation (46) in (Avram and Zhou 2016). Nevertheless, we provide an alternative proof in Appendix A.
