*5.1. No Transaction Cost*

In this subsection, we consider the problem in Equation (30) without transaction cost, i.e., *δ* = 0. For this case, we denote *<sup>V</sup>*(*<sup>x</sup>*, *K*) := *<sup>V</sup>*0(*<sup>x</sup>*, *K*) and *<sup>V</sup><sup>D</sup>*(*<sup>x</sup>*, *K*) := *<sup>V</sup>D*0 (*<sup>x</sup>*, *<sup>K</sup>*). From Section 3.2, recall that for each Λ ≥ 1, the optimal strategy is a barrier strategy, which is determined by *a*Λ defined in Equation (16), and its NPV satisfies *V*Λ = *v<sup>a</sup>*ΛΛ , where *v<sup>a</sup>*ΛΛ is as in Equation (13). Given a barrier strategy at *a* > 0 and *x* ∈ [0, *a*], the expected NPV of the injected capital is given by the function

$$\Psi\_x(a) := \mathbb{E}\_x \left[ \int\_0^\infty \mathbf{e}^{-qt} \, \mathrm{d}R\_t^{a,0} \right] = \frac{Z^{(q)}(a)}{q \mathcal{W}^{(q)}(a)} Z^{(q)}(\mathbf{x}) - k^{(q)}(\mathbf{x}), \tag{32}$$

with *k*(*q*) as in Equation (10). Clearly, if *x* > *a*, then <sup>Ψ</sup>*x*(*a*) = <sup>Ψ</sup>*a*(*a*). We also define

$$\underline{\mathbb{K}}\_{\mathbf{x}} := \lim\_{a \to \infty} \Psi\_{\mathbf{x}}(a). \tag{33}$$

Using Equation (12) and the properties of scale functions (see Remark 3 (3)),

$$
\underline{\mathbb{K}}\_{\mathfrak{x}} = -k^{(q)}(\mathfrak{x}) + \frac{Z^{(q)}(\mathfrak{x})}{\Phi(q)}.
$$

Note that *Kx* is the expected present value of the injected capital for the pay-nothing strategy *πPN* := {0, *<sup>R</sup>*<sup>0</sup>}. Therefore, letting *a* → ∞ in Equation (13), it can be verified

$$
v\_{\Lambda}^{\pi\_{\mathcal{PN}}}(x,\mathcal{K}) = \Lambda(\mathcal{K} - \underline{\mathbb{K}}\_x).
$$

Hence, if *K* ≥ *Kx*, then for any *x* ≥ 0,

$$V(\mathbf{x}, \mathcal{K}) = \sup\_{\pi \in \Theta} \inf\_{\Lambda \ge 0} v\_{\Lambda}^{\pi}(\mathbf{x}, \mathcal{K}) \ge \inf\_{\Lambda \ge 0} v\_{\Lambda}^{\pi p\_{\mathcal{N}}}(\mathbf{x}, \mathcal{K}) = 0. \tag{34}$$

Conversely, if *K* < *Kx*, the problem in Equation (30) is infeasible, which is verified below.

**Lemma 4.** *If K* < *Kx, then <sup>V</sup>*(*<sup>x</sup>*, *K*) = <sup>−</sup>∞*.*

**Proof.** First, by Remark 7 and Equation (11), it is easy to verify that

$$\lim\_{\Lambda \to \infty} \mathbb{E}\_{\mathbf{x}} \left[ \int\_0^\infty \mathbf{e}^{-qt} \, \mathbf{d} L\_t^{a\_\Lambda, 0} \right] = 0, \quad \text{for } \mathbf{x} \ge \mathbf{0}. \tag{35}$$

Then,

$$\begin{split} V^D(\mathbf{x}, K) &= \inf\_{\Lambda \ge 1} \left\{ \Lambda K + V\_{\Lambda}(\mathbf{x}) \right\} \\ &= \inf\_{\Lambda \ge 1} \left\{ \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathbf{d}L\_{t}^{a\_{\Lambda}, 0} \right] + \Lambda (K - \Psi\_{x}(a\_{\Lambda})) \right\} \\ &\leq \lim\_{\Lambda \to \infty} \left\{ \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathbf{d}L\_{t}^{a\_{\Lambda}, 0} \right] + \Lambda (K - \Psi\_{x}(a\_{\Lambda})) \right\} = -\infty. \end{split}$$

Now, since *<sup>V</sup>*(*<sup>x</sup>*, *K*) ≤ *<sup>V</sup><sup>D</sup>*(*<sup>x</sup>*, *K*) for any *x* ≥ 0, *K* ≥ 0, we have the result.

The next lemma allows us to prove that, when *K* = *Kx*, Equation (34) holds with equality, and it is used to prove the main result of this subsection.

**Lemma 5.** *Let x* ≥ 0 *be fixed. The function* Ψ*x is strictly decreasing on* (0, <sup>∞</sup>)*.*

**Proof.** First, consider the case when *x* < *a*. Then, by Remark 4 (i), we have that *qW*(*q*)(*a*) *<sup>Z</sup>*(*q*)(*a*) is strictly increasing and the lemma is obtained. Now, when *x* ≥ *a* > 0, a simple calculation shows that

$$\frac{d\Psi\_a(a)}{da} = -\frac{Z^{(q)}(a)\left(\mathcal{W}^{(q)\prime}(a)Z^{(q)}(a) - q[\mathcal{W}^{(q)}(a)]^2\right)}{q[\mathcal{W}^{(q)}(a)]^2} = -\frac{Z^{(q)}(a)\mathcal{W}^{(q)\prime}(a)}{q[\mathcal{W}^{(q)}(a)]^2}H(a),$$

which is strictly negative, by Remarks 3 and 5. From here, we conclude the assertion of the lemma.

**Lemma 6.** *If K* = *Kx, then <sup>V</sup>*(*<sup>x</sup>*, *K*) = 0 *and the optimal strategy is the pay-nothing strategy πPN.*

**Proof.** By Equation (34), we know that *<sup>V</sup>*(*<sup>x</sup>*, *K*) ≥ 0. On the other hand, from Lemma 5 and Equation (33), we have that Λ(*K* − <sup>Ψ</sup>*x*(*<sup>a</sup>*Λ)) ≤ 0 for all Λ ≥ 0. Then, using Equations (33) and (35)

$$\begin{split} V^D(\mathbf{x}, \mathbf{K}) = \inf\_{\Lambda \ge 1} \left\{ \Lambda \mathbf{K} + V\_{\Lambda}(\mathbf{x}) \right\} = \inf\_{\Lambda \ge 1} \left\{ \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathbf{d}L\_{t}^{a\_{\Lambda},0} \right] + \Lambda (\mathbf{K} - \mathbb{Y}\_{x}(a\_{\Lambda})) \right\} \\ \leq \lim\_{\Lambda \to \infty} \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathbf{d}L\_{t}^{a\_{\Lambda},0} \right] = 0. \quad \Box \end{split}$$

Now, we define

$$\overline{\mathbb{K}} := \lim\_{a \to 0} \Psi\_a(a).$$

Using Equation (12), we have that *K* = ∞ when the risk process has unbounded variation. Otherwise, by Remark 3 (2),

$$
\overline{X} = \frac{c - \psi'(0+)}{q},
\tag{36}
$$

and *K* corresponds to the expected NPV of the injected capital for the strategy *<sup>π</sup>*0,0 (see Equation (4.5) in Avram et al. (2007)).

**Lemma 7.** *Assume that the risk process X has bounded variation. If K* ≥ *K, then <sup>V</sup>*(*<sup>x</sup>*, *K*) = *K* + *<sup>V</sup>*1(*x*)*, with <sup>V</sup>*1(*x*) = *x* + *ψ*(0+) *q.*

**Proof.** If the Lévy measure is finite, by Equation (16), we have that *a*1 = 0. The same is true for the infinite Lévy measure case since *<sup>H</sup>*−<sup>1</sup>(1) = 0 by Remark 5. Using Equation (13) and Remark 6, we obtain

$$V\_1(\mathbf{x}) = v\_1^0(\mathbf{x}) = \mathbf{x} + \frac{\mathbf{c}}{q} - \mathbb{K} = \mathbf{x} + \frac{\psi'(0+)}{q}, \quad \text{for } \mathbf{x} \ge 0. \tag{37}$$

Now, by Equations (31), (36) and (37) and the weak duality, we ge<sup>t</sup>

$$V(\mathbf{x}, \mathbf{K}) \le V^D(\mathbf{x}, \mathbf{K}) \le K + v\_1^0(\mathbf{x}) = K + V\_1(\mathbf{x}).$$

Since *K* ≥ *K*, *<sup>π</sup>*0,0 is a feasible strategy. Then, using Equation (11), it yields,

$$V(\mathbf{x}, K) \ge \inf\_{\Lambda \ge 1} \{ v\_{\Lambda}^0(\mathbf{x}) + \Lambda K \} = \mathbf{x} + K - \frac{c - \psi'(0+)}{q} + \frac{c}{q} = K + V\_1(\mathbf{x}).$$

Therefore, *<sup>V</sup>*(*<sup>x</sup>*, *K*) = *K* + *<sup>V</sup>*1(*x*).

We are now ready for the main result of this subsection.

**Theorem 2.** *Assume δ* = 0 *and let V and V<sup>D</sup> as in Equation* (30) *and Equation* (31)*, respectively, then V* = *VD. Furthermore, if x and K are such that K* ∈ (*Kx*, *<sup>K</sup>*)*, then*

$$V(\mathbf{x}, \mathbf{K}) = \Lambda^\* \mathbf{K} + V\_{\Lambda^\*}(\mathbf{x}) = \mathbb{E}\_x \left[ \int\_0^\infty \mathbf{e}^{-qt} \, \mathbf{d}L\_t^{a^\*,0} \right],\tag{38}$$

*where a*<sup>∗</sup> = Ψ−<sup>1</sup> *x* (*K*)*, and* Λ∗ = 1 *<sup>H</sup>*(*a*<sup>∗</sup>)*.*

**Proof.** Lemmas 4, 6 and 7 show imply that Equation (38) holds when *x* and *K* are such that *K* ∈ [0, *Kx*] ∪ [*<sup>K</sup>*, <sup>∞</sup>). Assume now that *K* ∈ (*Kx*, *<sup>K</sup>*), then by Lemma 5 the function Ψ*x* is injective, so there exists a unique *a*<sup>∗</sup> > 0 such that <sup>Ψ</sup>*x*(*a*<sup>∗</sup>) = *K*. Note that from Equation (16), we have that there exists a unique Λ∗ such that *a*Λ∗ = *<sup>a</sup>*<sup>∗</sup>. Then,

$$\begin{split} V^{D}(\mathbf{x}, \mathbf{K}) &\leq \Lambda^{\*} \mathbf{K} + V\_{\Lambda^{\*}}(\mathbf{x}) \\ &= \Lambda^{\*} \mathbf{K} + \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathbf{d}L\_{t}^{a^{\*},0} \right] - \Lambda^{\*} \Psi\_{\mathbf{x}}(a^{\*}) \\ &= \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathbf{d}L\_{t}^{a^{\*},0} \right]. \end{split}$$

Meanwhile, since the strategy *<sup>π</sup>a*∗,<sup>0</sup> is feasible, we see

$$\begin{split} V(\mathbf{x}, K) \geq \inf\_{\Lambda \geq 1} \left\{ \boldsymbol{v}\_{\Lambda}^{\mathsf{T}\_{\mathbf{x}} \ast, 0}(\mathbf{x}) + \Lambda K \right\} = \inf\_{\Lambda \geq 1} \left\{ \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathrm{d}L\_{t}^{\mathbf{a}^{\star}, 0} \right] + \Lambda (K - \mathbb{Y}\_{\mathbf{x}}(\mathbf{a}^{\star})) \right\} \\ = \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathrm{d}L\_{t}^{\mathbf{a}^{\star}, 0} \right]. \end{split}$$

This implies that *<sup>V</sup><sup>D</sup>*(*<sup>x</sup>*, *K*) ≤ *<sup>V</sup>*(*<sup>x</sup>*, *<sup>K</sup>*). Finally, the weak duality gives Equation (46).
