**3. Preliminaries**

In this section, we revise the scale functions of spectrally negative Lévy processes and their properties (see, e.g., Kuznetsov et al. (2013); Kyprianou (2014)). We also recall well known results regarding optimal dividend strategies with capital injection for spectrally one-sided Lévy processes when the transaction cost is equal to 0 (i.e., *δ* = 0).

For each *q* ≥ 0, there exists a map *W*(*q*) : R −→ [0, <sup>∞</sup>), called *q*-scale function, satisfying *<sup>W</sup>*(*q*)(*x*) = 0 for *x* ∈ (−∞, <sup>0</sup>), and strictly increasing on [0, <sup>∞</sup>), which is defined by its Laplace transform:

$$\int\_0^\infty \mathbf{e}^{-\theta x} W^{(q)}(x) d\mathbf{x} = \frac{1}{\psi(\theta) - q'}, \quad \theta > \Phi(q), \tag{4}$$

where

$$\Phi(q) := \sup \{ \lambda \ge 0 : \psi(\lambda) = q \}.$$

We also define, for *x* ∈ R,

$$\begin{aligned} \overline{W}^{(q)}(\mathbf{x}) &:= \int\_0^\mathbf{x} \mathcal{W}^{(q)}(y) \mathbf{d}y, \quad Z^{(q)}(\mathbf{x}) := 1 + q \overline{\mathcal{W}}^{(q)}(\mathbf{x}),\\ \overline{Z}^{(q)}(\mathbf{x}) &:= \int\_0^\mathbf{x} Z^{(q)}(z) \mathbf{d}z = \mathbf{x} + q \int\_0^\mathbf{x} \int\_0^z \mathcal{W}^{(q)}(w) \mathbf{d}w \mathbf{d}z. \end{aligned}$$

Since *W*(*q*) is equal to zero on (−∞, <sup>0</sup>), we have

$$
\overline{W}^{(q)}(\mathbf{x}) = 0, \quad Z^{(q)}(\mathbf{x}) = 1 \quad \text{and} \quad \overline{Z}^{(q)}(\mathbf{x}) = \mathbf{x}, \quad \mathbf{x} \le 0.
$$
