**Appendix B. Proof of Proposition 2**

Recall from (7) that, for *x* ∈ (0, <sup>∞</sup>), we have

$$Z\_q'(\mathbf{x}, \Phi(p+q)) = p \int\_0^\infty \mathbf{e}^{-\Phi(p+q)y} \mathcal{W}^{(q)}(\mathbf{x}+y) d\mathbf{y}.\tag{A3}$$

By Theorem 1.2 in (Loeffen and Renaud 2010), if the tail of the Lévy measure is log-convex, then *W*(*q*) is log-convex. Using the properties of log-convex functions, as presented in (Roberts and Varberg 1973), we can deduce that *x* → *p*e<sup>−</sup><sup>Φ</sup>(*p*+*q*)*yW*(*q*)(*x* + *y*) is log-convex on (0, <sup>∞</sup>), for any fixed *y* ∈ (0, <sup>∞</sup>). Then, as Riemann integrals are limits of partial sums, we have that *x* → *<sup>Z</sup>q*(*<sup>x</sup>*, <sup>Φ</sup>(*p* + *q*)) is also a log-convex function on (0, <sup>∞</sup>). In particular, *<sup>Z</sup>q*(·, <sup>Φ</sup>(*p* + *q*)) is convex on (0, <sup>∞</sup>), so we can write, for some fixed *c* > 0,

$$Z\_q'(\mathbf{x}, \Phi(p+q)) = Z\_q'(\mathbf{c}, \Phi(p+q)) + \int\_{\mathcal{c}}^{\mathbf{x}} Z\_q'''(y, \Phi(p+q)) \mathbf{d}y.$$

where *<sup>Z</sup>*−*q* (·, <sup>Φ</sup>(*p* + *q*)) is the left-hand derivative of *<sup>Z</sup>q*(·, <sup>Φ</sup>(*p* + *q*)). Since *<sup>Z</sup>*−*q* (·, <sup>Φ</sup>(*p* + *q*)) is increasing and lim*x*→∞ *<sup>Z</sup>q*(*<sup>x</sup>*, <sup>Φ</sup>(*p* + *q*)) = <sup>∞</sup>, we have that the function *<sup>Z</sup>q*(·, <sup>Φ</sup>(*p* + *q*)) is ultimately strictly increasing. This proves that *b*∗*p* is well-defined.

It is known that *W*(*q*) is strictly increasing on (*b*<sup>∗</sup>∞, ∞); see (Loeffen and Renaud 2010). Then, using together the representations of *<sup>Z</sup>q*(*<sup>x</sup>*, <sup>Φ</sup>(*p* + *q*)) given in (5) and (7), we obtain

$$\begin{array}{rcl} Z\_q''(\mathbf{x}, \Phi(p+q)) &=& \Phi(p+q)p \int\_0^\infty \mathbf{e}^{-\Phi(p+q)y} \mathcal{W}^{(q)\prime}(\mathbf{x}+y) \mathrm{d}y - p \mathcal{W}^{(q)\prime}(\mathbf{x}) \\ &> p \mathcal{W}^{(q)\prime}(\mathbf{x}) \int\_0^\infty \Phi(p+q) \mathbf{e}^{-\Phi(p+q)y} \mathrm{d}y - p \mathcal{W}^{(q)\prime}(\mathbf{x}) = 0, \end{array} \tag{A4}$$

for all *x* > *b*<sup>∗</sup>∞. In other words, *x* → *<sup>Z</sup>q*(*<sup>x</sup>*, <sup>Φ</sup>(*p* + *q*)) is strictly increasing on (*b*<sup>∗</sup>∞, <sup>∞</sup>). Consequently, *b*∗*p* ≤ *b*<sup>∗</sup>∞.

The rest of the proof is similar to Lemma 3 in (Kyprianou et al. 2012), where a function closely related to one of the representations of *<sup>Z</sup>q*(*<sup>x</sup>*, <sup>Φ</sup>(*p* + *q*)) appears. For simplicity, set *g*(*x*) = *<sup>Z</sup>q*(*<sup>x</sup>*, <sup>Φ</sup>(*p* + *q*)). Using (5), we can write, for *x* > 0,

$$\mathbf{g}'(\mathbf{x}) = \Phi(p+q) \left( \mathbf{g}(\mathbf{x}) - \frac{p}{\Phi(p+q)} W^{(q)\prime}(\mathbf{x}) \right).$$

It follows that *g*(*x*) > 0 (resp. *g*(*x*) < 0) if and only if *g*(*x*) > *p* <sup>Φ</sup>(*p* + *q*)*<sup>W</sup>*(*q*)(*x*) (resp. *g*(*x*) < *p* <sup>Φ</sup>(*p* + *q*)*<sup>W</sup>*(*q*)(*x*)). This means *g*(*b*) > *p* <sup>Φ</sup>(*p* + *q*)*<sup>W</sup>*(*q*)(*b*) for *b* < *b*∗*p* and *g*(*b*) < *p* <sup>Φ</sup>(*p* + *q*)*<sup>W</sup>*(*q*)(*b*) for *b* > *b*∗*p*. If *b*∗*p* > 0 then *<sup>g</sup>*(*b*<sup>∗</sup>*p*)=(*p*/Φ(*<sup>p</sup>* + *<sup>q</sup>*))*W*(*q*)(*b*<sup>∗</sup>*p*).

We deduce that *b*∗*p* > 0 if and only if *g*(0+) < (*p*/Φ(*p* + *<sup>q</sup>*))*W*(*q*)(0+), where *g*(0+) = <sup>Φ</sup>(*p* + *q*) − *<sup>p</sup>W*(*q*)(0). Written differently, we have *b*∗*p* > 0 if and only if

$$
\Phi(p+q) - pW^{(q)}(0) < \frac{p}{\Phi(p+q)} W^{(q)\prime}(0+).
$$

If *σ* > 0, then *W*(*q*)(0) = 0 and *W*(*q*)(0+) = 2/*σ*2, which implies that *b*∗*p* > 0 if and only if

$$\frac{\left(\Phi(p+q)\right)^2}{p} < \frac{2}{\sigma^2}.$$

If *σ* = 0 and *ν*(0, ∞) = <sup>∞</sup>, then *W*(*q*)(0+) = <sup>∞</sup>, which implies that *b*∗*p* > 0. Finally, if *σ* = 0 and *ν*(0, ∞) < <sup>∞</sup>, then *W*(*q*)(0) = 1/*<sup>c</sup>*, where *c* > 0 is the drift, and *W*(*q*)(0+) = (*q* + *ν*(0, ∞))/*c*2, which implies that *b*∗*p*> 0 if and only if

$$
\Phi(p+q) - \frac{p}{c} < \frac{p}{\Phi(p+q)} \frac{q + \nu(0, \infty)}{c^2}.
$$
