*3.1. Reflected Lévy Processes*

Let *S* = {*St* : *t* ≥ 0} and *R*<sup>0</sup> = {*R*0*t* : *t* ≥ 0} be defined, respectively, as

$$S\_t := \sup\_{0 \le s \le t} (X\_s \lor 0) \quad \text{and} \quad R\_t^0 := \sup\_{0 \le s \le t} (-X\_s \lor 0). \tag{6}$$

We denote *Y* ˆ := *S* − *X* and *Y* := *X* + *R*0, which are strong Markov processes. Observe that the process *R*<sup>0</sup> pushes *X* upwards whenever it attempts to down-cross the level 0; as a result the process *Y* only takes values on [0, <sup>∞</sup>). An introduction to the theory of Lévy processes and their reflected processes can be encountered in Bertoin (1998); Kyprianou (2014).

Let *τ*ˆ*a* be defined as *τ*ˆ*a* = inf{*t* > 0 : *Y*ˆ*t* ∈ (*a*, <sup>∞</sup>)}, with *a* > 0. Then, by Proposition 2 in Pistorius (2004),

$$\mathbb{E}\_{-\mathbf{x}}\left[\mathbf{e}^{-q\mathbf{f}\_{\mathbf{z}}}\right] = Z^{(q)}(a-\mathbf{x}) - q\mathcal{W}^{(q)}(a-\mathbf{x})\frac{\mathcal{W}^{(q)}(a)}{\mathcal{W}^{(q)\prime}(a)}, \quad \mathbf{x} \in [0, a].$$

We define for *a* > 0,

$$H(a) := \mathbb{E}\_0\left[\mathbf{e}^{-q\mathbf{t}\_x}\right] = Z^{(q)}(a) - q \frac{[\mathcal{W}^{(q)}(a)]^2}{\mathcal{W}^{(q)\prime}(a)}.\tag{7}$$

**Remark 5.** *Note that, by definition, the function H is strictly positive, strictly decreasing and satisfies*

$$\lim\_{a \to \infty} H(a) = 0, \qquad \lim\_{a \to 0} H(a) = 1 - \frac{q[\mathcal{W}^{(q)}(0)]^2}{\mathcal{W}^{(q)\prime}(0+)}.$$

*Therefore, the function H has an inverse from* (0, 1 − *q*/(*q* + Π(0, ∞))) *onto* (0, ∞) *when σ* = 0 *and* Π(0, ∞) < <sup>∞</sup>*, and from* (0, 1) *onto* (0, ∞) *otherwise.*

Similarly, taking *κb* := inf{*t* > 0 : *Yt* ∈ (*b*, <sup>∞</sup>)}, with *b* > 0, we know from Proposition 2 in Pistorius (2004) that

$$\mathbb{E}\_{\mathbf{x}}\left[\mathbf{e}^{-q\chi\_{\mathbf{p}}}\right] = \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(b)}, \quad \mathbf{x} \le b. \tag{8}$$

.

In addition, we know from (Avram et al. 2007, page 167) that

$$\begin{split} \mathbb{E}\_{\mathbf{x}} \left[ \int\_{[0,\mathbf{x}\_{b}]} \mathbf{e}^{-qt} \, \mathrm{d}\mathbf{R}\_{t}^{0} \right] &= -\mathbb{Z}^{(q)}(\mathbf{x}) + \Phi(q)^{-1} \mathcal{Z}^{(q)}(\mathbf{x}) - \frac{\Psi'(0+)}{q} \\ &+ \left( \mathbb{Z}^{(q)}(b) - \Phi(q)^{-1} \mathcal{Z}^{(q)}(b) + \frac{\Psi'(0+)}{q} \right) \frac{\mathcal{Z}^{(q)}(\mathbf{x})}{\mathcal{Z}^{(q)}(b)} \\ &= -k^{(q)}(\mathbf{x}) + \frac{\mathcal{Z}^{(q)}(\mathbf{x})}{\mathcal{Z}^{(q)}(b)} k^{(q)}(b), \quad \mathbf{x} \le b, \end{split} \tag{9}$$

where

$$k^{(q)}(\mathbf{x}) := \mathbb{Z}^{(q)}(\mathbf{x}) + \frac{\Psi'(0+)}{q}. \tag{10}$$

### *3.2. Optimal Dividends without Transaction Cost and with Capital Injection*

When *δ* = 0, Equation (2) becomes

$$v\_{\Lambda}^{\pi}(\mathbf{x}) := v\_{0,\Lambda}^{\pi}(\mathbf{x}) = \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathbf{d}L\_{t}^{\pi} - \Lambda \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathbf{d}R\_{t}^{\pi} \right],$$

for any initial capital *x* ≥ 0 and admissible policy *π* = {*L<sup>π</sup>*, *Rπ*} ∈ Θ. Consider the strategy *<sup>π</sup>a*,<sup>0</sup> = {*La*,0, *Ra*,<sup>0</sup>}, which consists in setting reflecting barriers at *a* and 0, respectively. The controlled risk process *X<sup>π</sup>a*,<sup>0</sup> = *X* − *La*,<sup>0</sup> + *Ra*,<sup>0</sup> is a doubly reflected spectrally negative Lévy process and was studied by Avram et al. (2007). Intuitively, the process behaves similar to a Lévy process when it is inside [0, *a*], but when it tries to cross above the level *a* or below the level 0 it is forced to stay inside [0, *a*]. Using Theorem 1 from Avram et al. (2007), we have that for *a* > 0 and *x* ∈ [0, *a*],

$$\mathbb{E}\_x \left[ \int\_0^\infty e^{-qt} \, \mathrm{d}L\_t^{a,0} \right] = \frac{Z^{(q)}(\mathbf{x})}{q \mathcal{W}^{(q)}(a)},\tag{11}$$

$$\mathbb{E}\_x \left[ \int\_0^\infty \mathbf{e}^{-qt} \, \mathrm{d}R\_t^{a,0} \right] = \frac{Z^{(q)}(a)}{q \mathcal{W}^{(q)}(a)} Z^{(q)}(\mathbf{x}) - k^{(q)}(\mathbf{x}). \tag{12}$$

Note that the expression in Equation (12) is finite under our assumption that *ψ* (0+) > − ∞. Using the expressions above, we can see that, for Λ ≥ 1,

$$v^{a}\_{\Lambda}(\mathbf{x}) := v^{\pi\_{a0}}\_{\Lambda}(\mathbf{x}) = \begin{cases} Z^{(q)}(\mathbf{x})\mathbb{I}\_{\Lambda}(a) + \Lambda k^{(q)}(\mathbf{x}), & \text{if } 0 \le \mathbf{x} \le a, \\\mathbf{x} - a + v^{a}\_{\Lambda}(a), & \text{if } \mathbf{x} > a, \end{cases} \tag{13}$$

where

$$\zeta\_{\Lambda}(a) := \frac{1 - \Lambda Z^{(q)}(a)}{q \mathcal{W}^{(q)}(a)}, \quad a > 0. \tag{14}$$

Equation (13) suggests that, to find the best barrier strategy we should maximize the function *ζ*<sup>Λ</sup>. Thus, we can define the candidate for the optimal barrier by

$$a\_{\Lambda} = \sup \{ a \ge 0 : \mathbb{Z}\_{\Lambda}(a) \ge \mathbb{Z}\_{\Lambda}(\mathbf{x}), \text{ for all } \mathbf{x} \ge \mathbf{0} \}. \tag{15}$$

**Remark 6.** *Note that ζ*Λ : (0, ∞) −→ (− <sup>∞</sup>, 0) *and satisfies*

$$\lim\_{a \to 0} \zeta\_{\Lambda}(a) = -\frac{\Lambda - 1}{q \mathcal{W}^{(q)}(0)} \quad \text{and} \quad \lim\_{a \to \infty} \zeta\_{\Lambda}(a) = -\frac{\Lambda}{\Phi(q)}.$$

*Here, in case that X is of unbounded variation, the first equality is understood to be* <sup>−</sup>∞*. The barrier level a*Λ*, given in Equation* (15)*, corresponds with the level defined in Avram et al. (2007). Using the definition of the function H, we have that*

$$\frac{\mathrm{d}\mathbb{Z}\_{\Lambda}(a)}{\mathrm{d}a} = \frac{\Lambda \mathcal{W}^{(q)\prime}(a)}{q[\mathcal{W}^{(q)}(a)]^2} (H(a) - 1/\Lambda).$$

*Since H is strictly decreasing, ζ*Λ *has a unique maximum at a*Λ *that is either a critical point, which is a solution of <sup>H</sup>*(*a*) = 1Λ*, or* 0 *if the right-hand derivative of ζ*Λ *is negative at* 0*. Therefore, by Remark 5,*

$$a\_{\Lambda} = \begin{cases} 0, & \text{if } \sigma = 0, \ \Pi(0, \infty) < \infty \text{ and } \Lambda < 1 + \frac{q}{\Pi(0, \infty)}, \\\\ H^{-1} \left( 1/\Lambda \right), & \text{otherwise.} \end{cases} \tag{16}$$

*In addition, note that ζ*Λ *is strictly increasing on* (0, *<sup>a</sup>*Λ) *and strictly decreasing on* (*<sup>a</sup>*Λ, <sup>∞</sup>)*.*

Hence, from Avram et al. (2007), we know that the value function in Equation (3) and the optimal strategy are given by *V*Λ := *<sup>V</sup>*0,Λ = *v<sup>a</sup>*ΛΛ and *<sup>π</sup>*0,*<sup>a</sup>*Λ , where *v<sup>a</sup>*ΛΛ and *a*Λ are as in Equations (13) and (16), respectively.

**Remark 7.** *Note that the optimal barrier a*Λ → ∞ *as* Λ → ∞*.*

### **4. Capital Injection and Fixed Transaction Cost**

In this section, we solve the problem in Equation (3) in the presence of a fixed transaction cost *δ* > 0. We consider strategies where the capital injection policy is *R*0, given in Equation (6), and the dividend strategy is the so-called reflected (*<sup>c</sup>*1, *<sup>c</sup>*2)-policy, defined below.

### *4.1. Value Function of Reflected* (*<sup>c</sup>*1, *<sup>c</sup>*2)*-Policies*

Let (*<sup>c</sup>*1, *<sup>c</sup>*2) be a pair such that 0 ≤ *c*1 < *c*2. In this subsection, we define the reflected (*<sup>c</sup>*1, *<sup>c</sup>*2)-policy, denoted by *<sup>π</sup>*(*<sup>c</sup>*1,*c*2),0, and under which we construct the controlled process. Let *Y* = *X* + *R*<sup>0</sup> be the Lévy process reflected from below 0, so we set

$$X\_t^{(c\_1, c\_2), 0} = Y\_{t\_\prime} \quad \text{for } t \le T\_1^{c\_1, c\_2}.$$

where *T<sup>c</sup>*1,*c*<sup>2</sup> 1 = inf{*t* > 0 : *Yt* > *<sup>c</sup>*2}. The process then jumps downward by *YTc*1,*c*2 1 − *c*1 so that *<sup>X</sup>*(*<sup>c</sup>*1,*c*2),<sup>0</sup> *T<sup>c</sup>*1,*c*<sup>2</sup> 1 = *c*1. Now, for *T<sup>c</sup>*1,*c*<sup>2</sup> 1 ≤ *t* < *T<sup>c</sup>*1,*c*<sup>2</sup> 2 = inf{*t* > *T<sup>c</sup>*1,*c*<sup>2</sup> 1 : *<sup>X</sup>*(*<sup>c</sup>*1,*c*2),<sup>0</sup> *t* > *<sup>c</sup>*2}, *X*(*<sup>c</sup>*1,*c*2),<sup>0</sup> is the reflected process from below at 0 of *Xt* + (*<sup>c</sup>*1 − *XTc*1,*c*2 1 ), and *<sup>X</sup>*(*<sup>c</sup>*1,*c*2),<sup>0</sup> *T<sup>c</sup>*1,*c*<sup>2</sup> 2 = *c*1. By repeating this procedure, we can construct the process inductively. The process *X*(*<sup>c</sup>*1,*c*2),<sup>0</sup> clearly admits the decomposition

$$X\_t^{(c\_1, c\_2), 0} = X\_t - L\_t^{(c\_1, c\_2), 0} + R\_t^{(c\_1, c\_2), 0}, \quad t \ge 0, 1$$

where *L*(*<sup>c</sup>*1,*c*2),<sup>0</sup> and *R*(*<sup>c</sup>*1,*c*2),<sup>0</sup> are the cumulative amounts of dividend payments and capital injection, respectively.

Let us compute the expected NPV of dividends with transaction costs for this strategy. For this purpose, we denote

$$f\_{\mathcal{E}\_1, \mathcal{E}\_2}(\mathbf{x}) = \mathbb{E}\_{\mathcal{X}} \left[ \int\_0^\infty \mathbf{e}^{-qt} \, \mathbf{d} \left( L\_t^{(c\_1, c\_2), 0} - \delta \sum\_{0 \le s \le t} \mathbf{1}\_{\left\{ \Delta L\_s^{(c\_1, c\_2), 0} > 0 \right\}} \right) \right] \right].$$

If *x* < *c*2, by the Strong Markov Property and Equation (8), we obtain that

$$f\_{\mathcal{E}\_1 \mathcal{E}\_2}(\mathbf{x}) = \mathbb{E}\_{\mathcal{X}} \left[ \mathbf{e}^{-q \, T\_1^{\mathcal{E}\_1 \mathcal{E}\_2}} \right] f\_{\mathcal{E}\_1 \mathcal{E}\_2}(\mathbf{c}\_2) = \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(\mathbf{c}\_2)} f\_{\mathcal{E}\_1 \mathcal{E}\_2}(\mathbf{c}\_2). \tag{17}$$

When *x* ≥ *c*2, an amount *x* − *c*1 is paid as dividends and a transaction cost *δ* is incurred immediately, so by using Equation (17) we obtain

$$f\_{\mathbb{C}\_1\mathbb{C}\_2}(\mathbf{x}) = \mathbf{x} - \mathbf{c}\_1 - \delta + f\_{\mathbb{C}\_1\mathbb{C}\_2}(\mathbf{c}\_1) = \mathbf{x} - \mathbf{c}\_1 - \delta + \frac{Z^{(q)}(\mathbf{c}\_1)}{Z^{(q)}(\mathbf{c}\_2)} f\_{\mathbb{C}\_1\mathbb{C}\_2}(\mathbf{c}\_2).$$

Hence, taking *x* = *c*2, and solving for *fc*1,*c*2 (*<sup>c</sup>*2) we ge<sup>t</sup>

$$f\_{\mathfrak{c}1,\mathfrak{c}2}(\mathfrak{c}\_2) = (\mathfrak{c}\_2 - \mathfrak{c}\_1 - \delta) \frac{Z^{(q)}(\mathfrak{c}\_2)}{Z^{(q)}(\mathfrak{c}\_2) - Z^{(q)}(\mathfrak{c}\_1)}.$$

Using the aforementioned expression in Equation (17), we have for *x* < *c*2,

$$f\_{\mathfrak{c}1,\mathfrak{c}2}(\mathbf{x}) = (\mathfrak{c}\_2 - \mathfrak{c}\_1 - \delta) \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(\mathfrak{c}\_2) - Z^{(q)}(\mathfrak{c}\_1)}.\tag{18}$$

Now, let us calculate the expected NPV of the injected capital denoted by

$$\mathcal{g}\_{\mathfrak{c}\_1,\mathfrak{c}\_2}(\mathfrak{x}) = \mathbb{E}\_{\mathfrak{x}} \left[ \int\_0^\infty \mathbf{e}^{-qt} \, \mathrm{d}R\_t^{(\mathfrak{c}\_1,\mathfrak{c}\_2),0} \right] \cdot \mathfrak{x}$$

Again, by the Strong Markov Property, noting that *T<sup>c</sup>*1,*c*<sup>2</sup> 1 = inf{*t* > 0 : *Yt* ∈ (*<sup>c</sup>*2, ∞)} and Equations (8)–(9), we have for *x* ≥ 0

$$\begin{split} \mathcal{g}\_{\mathcal{E}\_{1}\mathcal{E}\_{2}}(\mathbf{x}) &= \mathbb{E}\_{\mathbf{x}} \left[ \int\_{[0,T\_{1}^{\mathcal{E}\_{1}\mathcal{E}\_{2}}]} \mathbf{e}^{-qt} \, \mathbf{d} \mathcal{R}\_{t}^{0} \right] + \mathbb{E}\_{\mathbf{x}} \left[ \mathbf{e}^{-q \, T\_{1}^{\mathcal{E}\_{1}\mathcal{E}\_{2}}} \right] \mathcal{g}\_{\mathcal{E}\_{1}\mathcal{E}\_{2}}(\mathbf{c}\_{1}) \\ &= -k^{(q)}(\mathbf{x}) + k^{(q)}(\mathbf{c}\_{2}) \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(\mathbf{c}\_{2})} + \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(\mathbf{c}\_{2})} \mathbf{g}\_{\mathcal{E}\_{1}\mathcal{E}\_{2}}(\mathbf{c}\_{1}). \end{split}$$

Thus, setting *x* = *c*1 and solving for *gc*1,*c*2 (*<sup>c</sup>*1), we obtain

$$\begin{split} \mathfrak{g}\_{\mathfrak{c}\_{1}\mathfrak{c}\_{2}}(\mathfrak{c}\_{1}) &= \left( -k^{(q)}(\mathfrak{c}\_{1}) + k^{(q)}(\mathfrak{c}\_{2}) \frac{Z^{(q)}(\mathfrak{c}\_{1})}{Z^{(q)}(\mathfrak{c}\_{2})} \right) \frac{Z^{(q)}(\mathfrak{c}\_{2})}{Z^{(q)}(\mathfrak{c}\_{2}) - Z^{(q)}(\mathfrak{c}\_{1})} \\ &= \left( -\overline{Z}^{(q)}(\mathfrak{c}\_{1}) + \overline{Z}^{(q)}(\mathfrak{c}\_{2}) \frac{Z^{(q)}(\mathfrak{c}\_{1})}{Z^{(q)}(\mathfrak{c}\_{2})} \right) \frac{Z^{(q)}(\mathfrak{c}\_{2})}{Z^{(q)}(\mathfrak{c}\_{2}) - Z^{(q)}(\mathfrak{c}\_{1})} - \frac{\mathfrak{y}^{\prime}(0 +)}{q} . \end{split}$$

Putting the pieces together, we obtain

$$\begin{split} g\_{\mathsf{C}1\cdot\mathsf{Z}}(\mathbf{x}) &= -k^{(q)}(\mathbf{x}) + k^{(q)}(\mathbf{c}\_{2}) \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(\mathbf{c}\_{2})} \\ &+ \left( \left( -\overline{Z}^{(q)}(\mathbf{c}\_{1}) + \overline{Z}^{(q)}(\mathbf{c}\_{2}) \frac{Z^{(q)}(\mathbf{c}\_{1})}{Z^{(q)}(\mathbf{c}\_{2})} \right) \frac{Z^{(q)}(\mathbf{c}\_{2})}{Z^{(q)}(\mathbf{c}\_{2}) - Z^{(q)}(\mathbf{c}\_{1})} - \frac{\Psi'(\mathbf{0}+)}{q} \right) \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(\mathbf{c}\_{2})} \\ &= -k^{(q)}(\mathbf{x}) + \overline{Z}^{(q)}(\mathbf{c}\_{2}) \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(\mathbf{c}\_{2})} + \left( -\overline{Z}^{(q)}(\mathbf{c}\_{1}) + \overline{Z}^{(q)}(\mathbf{c}\_{2}) \frac{Z^{(q)}(\mathbf{c}\_{1})}{Z^{(q)}(\mathbf{c}\_{2})} \right) \frac{Z^{(q)}(\mathbf{x})}{Z^{(q)}(\mathbf{c}\_{2}) - Z^{(q)}(\mathbf{c}\_{1})} \\ &= Z^{(q)}(\mathbf{x}) \left( \frac{\overline{Z}^{(q)}(\mathbf{c}\_{2}) - \overline{Z}^{(q)}(\mathbf{c}\_{1})}{Z^{(q)}(\mathbf{c}\_{2}) - Z^{(q)}(\mathbf{c}\_{1})} \right) - k^{(q)}(\mathbf{x}). \end{split}$$

> Hence, we have the following result.

**Lemma 1.** *The expected NPV associated with a reflected* (*<sup>c</sup>*1, *<sup>c</sup>*2)*-policy is given by*

$$v^{\varepsilon\_1,\varepsilon\_2}\_{\delta,\Lambda}(\mathbf{x}) := v^{\pi\_{\{\varepsilon\_1,\varepsilon\_2\},0}}\_{\delta,\Lambda}(\mathbf{x}) = \begin{cases} Z^{(q)}(\mathbf{x}) \mathbf{G}\_{\Lambda}(\mathbf{c}\_1,\mathbf{c}\_2) + \Lambda k^{(q)}(\mathbf{x}), & \text{if } \mathbf{x} \le \mathbf{c}\_2 \omega \\ \mathbf{x} - \mathbf{c}\_1 - \delta + v^{\varepsilon\_1,\varepsilon\_2}\_{\delta,\Lambda}(\mathbf{c}\_1), & \text{if } \mathbf{x} > \mathbf{c}\_2 \omega \end{cases}$$

*where*

$$G\_{\Lambda}(c\_1, c\_2) := \frac{c\_2 - c\_1 - \delta - \Lambda \left( \overline{Z}^{(q)}(c\_2) - \overline{Z}^{(q)}(c\_1) \right)}{Z^{(q)}(c\_2) - Z^{(q)}(c\_1)}, \quad \text{for all } c\_2 > c\_1 \ge 0. \tag{19}$$

**Remark 8.** *Note that G*Λ *is* C<sup>2</sup> *on* A := {(*<sup>c</sup>*1, *<sup>c</sup>*2) ∈ <sup>R</sup>2+ : *c*1 < *<sup>c</sup>*2}*, and*

$$\lim\_{\varepsilon\_2 \downarrow \varepsilon\_1} G\_\Lambda(\varepsilon\_1, \varepsilon\_2) = -\infty,\text{ for } \varepsilon\_1 \ge 0 \text{ fixed},$$

$$\lim\_{|\varepsilon\_1| + |\varepsilon\_2| \to \infty} G\_\Lambda(\varepsilon\_1, \varepsilon\_2) = \lim\_{\varepsilon\_2 \to \infty} G\_\Lambda(\varepsilon\_1, \varepsilon\_2) = -\frac{\Lambda}{\Phi(q)}.$$
