*1.2. The Problem*

For the stochastic control problem considered in this paper, an admissible strategy is represented by a pair (*<sup>C</sup>*, *D*) composed of a non-decreasing, left-continuous, and adapted stochastic process *D* = {*Dt*, *t* ≥ 0} and *C* = {*Ct*, *t* ≥ <sup>0</sup>}, where *Dt* represents the cumulative amount of dividends paid up to time *t*, while *Ct* represents the cumulative amount of capital injections made up to time *t*. We assume *D*0 = 0 and *C*0 = 0. For a given strategy (*<sup>C</sup>*, *<sup>D</sup>*), the corresponding controlled surplus process *U* = {*Ut*, *t* ≥ 0} is defined by *Ut* = *Xt* − *Dt* + *Ct*. Define also *τ* = inf {*t* > 0: *Ut* < <sup>0</sup>}.

For a given initial surplus *x* ≥ 0, let A(*x*) be the corresponding set of admissible strategies. Also, let *q* > 0 be the discounting rate and let *k* > 1 be the proportional cost of injecting capital. The objective is to maximize the value of a strategy using the following objective function:

$$J(\mathbf{x}, \mathbf{C}, D) = \mathbb{E}\_{\mathbf{x}} \left[ \int\_0^\pi \mathbf{e}^{-qt} \left( \mathbf{d}D\_t - k \mathbf{d} \mathbf{C}\_t \right) \right],\tag{3}$$

that is, the goal is to find the optimal value function

$$V\_k(\mathbf{x}) = \sup\_{(\mathsf{C},D)\in \mathcal{A}(\mathbf{x})} J(\mathbf{x}, \mathsf{C}, D).$$

For a general Markov process *X*, our problem amounts to solving (in a viscosity sense) the following Hamilton–Jacobi–Bellman (HJB) equation :

$$\begin{cases} \max\left\{ \left( \mathcal{L} - q \right) V(\mathbf{x}), 1 - V'(\mathbf{x}), V'(\mathbf{x}) - k, -V(\mathbf{x}) \right\} \le 0, & \mathbf{x} \ge 0\\ \max\left\{ \left( \mathcal{L} - q \right) V(\mathbf{x}), V'(\mathbf{x}) - k, -V(\mathbf{x}) \right\} \le 0, & \mathbf{x} < 0 \end{cases} \tag{4}$$

where L is the infinitesimal generator associated with the underlying uncontrolled process *X* (see also (Zhu and Yang 2016, sct. 3.6) for the case of diffusions). For the Cramér-Lundberg process with exponential jumps, the operator is

$$
\mathcal{L}V(\mathbf{x}) = cV'(\mathbf{x}) + \lambda\mu \int\_0^\infty \left(V(\mathbf{x} - z) - V(\mathbf{x})\right) \mathbf{e}^{-\mu z} d\mathbf{z}.\tag{5}
$$

The second part of (4) is associated to the possibility of modifying the surplus by a lump sum dividend paymen<sup>t</sup> (see (6) below), and the third part to capital injections.

In the cases already studied, the Løkka–Zervos alternative reduces to the following dilemma: shall we declare bankruptcy at level 0, or shall we use capital injections to maintain the surplus positive?

The classical problems studied by (de Finetti 1957; Shreve et al. 1984) are revisited in Section 2 and new results are obtained. In Section 3, we prove that the Løkka–Zervos alternative holds for a Cramér–Lundberg model with exponential jumps.

### **2. The Classical Dividend Problems for SNLPs**

In this section, we review de Finetti's, as well as Shreve, Lehoczky, and Gaver's optimal dividend problems for general spectrally negative Lévy processes. As is well-known, the value functions can be expressed in terms of scale functions (see, for example, Avram et al. 2004, 2019; Bertoin 1998; Kyprianou 2014).
