**2. Formulation of the Problem**

Let *X* = {*Xt* : *t* ≥ 0} be a Lévy process defined on a probability space (<sup>Ω</sup>, F, <sup>P</sup>), and let F := {F*t* : *t* ≥ 0} be the completed and right-continuous filtration generated by *X*. Recall that a Lévy process is a process that has càdlàg paths and stationary and independent increments. For *x* ∈ R, we denote by P*x* the law of *X*, where *X*0 = *x*. For convenience, we take P0 ≡ P, when *x* = 0. The expectation operator associated with P*x* is denoted by E*<sup>x</sup>*. We take E0 ≡ E, where E is the expectation operator associated with P.

We henceforth assume that the insurance company's surplus *X* is modeled by a spectrally negative process, i.e., a Lévy process that only has negative jumps. We omit the case when *X* has monotone trajectories to avoid trivial cases.

The Laplace exponent of *X* is given by

$$\psi(\theta) := \log \mathbb{E}[\theta X\_1] = \gamma \theta + \frac{\sigma^2}{2} \theta^2 - \int\_{(0,\infty)} \left(1 - \mathbf{e}^{-\theta z} - \theta z \mathbf{1}\_{\{0 < z \le 1\}}\right) \Pi(\mathbf{d}z), \quad \theta \ge 0,$$

where *γ* ∈ R, *σ* ≥ 0, and the Lévy measure of *X*, Π, is a measure defined on (0, ∞) satisfying

$$\int\_{(0,\infty)} (1 \wedge z^2) \Pi(\mathbf{dz}) < \infty.$$

As is well-known, the process *X* has bounded variation paths if and only if *σ* = 0 and (0,1]*z*<sup>Π</sup>(d*z*) < ∞. In this case, *X* can be written as

$$X\_t = ct - \check{S}\_t, \qquad t \ge 0,\tag{1}$$

where *c* := *γ* + (0,1] *z*<sup>Π</sup>(d*z*) and *S* = {*<sup>S</sup>t* : *t* ≥ 0} is a drift-less subordinator. Since we omit the case when *X* has monotone paths, it is necessary that the constant *c* is greater than zero. Note that the Laplace exponent of *X*, with *X* as in Equation (1), is given as follows,

$$
\psi(\theta) = c\theta - \int\_{(0,\infty)} (1 - \mathbf{e}^{-\theta z}) \Pi(\mathbf{d}z), \quad \theta \ge 0.
$$

*De Finetti's Problem with Fixed Transaction Cost and Capital Injection*

Let *π* = {*L<sup>π</sup>*, *Rπ*} be a strategy, where *Lπ* is left-continuous P*x*-a.s., and *Rπ* is right-continuous P*x*-a.s. Additionally, we assume that *Lπ* and *Rπ* are non-negative, and non-decreasing P*x*-a.s., start at zero and are adapted to the filtration F. Then, the controlled process, *X<sup>π</sup>*, associated with the strategy *π*, is the following

$$X\_t^{\pi} = X\_t - L\_t^{\pi} + R\_t^{\pi}, \quad t \ge 0.$$

For each *t* ≥ 0, the quantities *<sup>L</sup><sup>π</sup>t* and *<sup>R</sup><sup>π</sup>t* represent the cumulative amounts that the insurance company has paid to its shareholders and has injected, respectively.

The set of admissible policies Θ consists of those policies *π* for which *Xπ* is non-negative and for *x* ≥ 0,

$$\mathbb{E}\_{\chi} \left[ \int\_0^{\infty} \mathbf{e}^{-qt} \, \mathbf{d} \mathcal{R}\_t^{\pi} \right] < \infty.$$

When there is a fixed transaction cost *δ* > 0, we only consider the class of admissible strategies *π* = {*L<sup>π</sup>*, *Rπ*} ∈ Θ such that

$$L\_t^\pi = \sum\_{0 \le s \le t} \Delta L\_s^\pi \quad \text{ } t \ge 0\text{ }.$$

where <sup>Δ</sup>*L<sup>π</sup>t*:= *<sup>L</sup><sup>π</sup>t*+ − *<sup>L</sup><sup>π</sup>t*. We denote this class by Θ*δ* and in the case *δ* = 0, we take Θ0 ≡ Θ.

Given an initial capital *x* ≥ 0 and a policy *π* = {*L<sup>π</sup>*, *Rπ*} ∈ Θ*δ*, with *δ* ≥ 0, we define the expected NPV as follows,

$$w^{\pi}\_{\delta,\Lambda}(\mathbf{x}) := \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathrm{d} \left( L\_{t}^{\pi} - \delta \sum\_{0 \le s \le t} \mathbf{1}\_{\{\Delta L\_{s}^{\pi} > 0\}} \right) - \Lambda \int\_{0}^{\infty} \mathbf{e}^{-qt} \, \mathrm{d}R\_{t}^{\pi} \right], \tag{2}$$

where *q* > 0, *δ* ≥ 0, and Λ > 0 is the unit cost per capital injected.

**Remark 1.** *Note that in the case of proportional transaction cost the expected NPV changes to*

$$\mathbb{E}\_{\mathbf{x}}\left[\int\_{0}^{\infty}\mathbf{e}^{-qt}\,\mathbf{d}\left(\beta L\_{t}^{\pi}-\delta\sum\_{0\le s\le t}\mathbf{1}\_{\{\Delta L\_{s}^{\pi}>0\}}\right)-\Lambda\int\_{0}^{\infty}\mathbf{e}^{-qt}\,\mathbf{d}R\_{t}^{\pi}\right],$$

*where* 0 < *β* < 1*, so by changing δ and* Λ *appropriately we can recover Equation* (2)*.*

Hence, the value function we aim to find is

$$V\_{\delta,\Lambda}(\mathbf{x}) := \sup\_{\pi \in \Theta\_{\delta}} v\_{\delta,\Lambda}^{\pi}(\mathbf{x}). \tag{3}$$

**Remark 2.** *Since we want to avoid this function taking the value* <sup>−</sup>∞*, we assume that ψ*(0+) = <sup>E</sup>[*<sup>X</sup>*1] > <sup>−</sup>∞*. We also assume that* Λ ≥ 1*, otherwise the value function will go to infinity since large amounts of dividends will be paid, given that the company will inject capital at a cheaper cost to bail out.*

Note that the problem in Equation (3) was studied by Avram et al. (2007) under the assumption *δ* = 0 (see Section 3.2). Therefore, we focus on the optimal control problem when *δ* > 0 (see Section 4).
