*1.1. Problem Formulation*

On a filtered probability space (<sup>Ω</sup>, F, {F*<sup>t</sup>*, *t* ≥ 0} , <sup>P</sup>), let *X* = {*Xt*, *t* ≥ 0} be a spectrally negative Lévy process with Laplace exponent *θ* → *ψ*(*θ*) and with *q*-scale functions *<sup>W</sup>*(*q*), *q* ≥ 0 given by

$$\int\_0^\infty \mathbf{e}^{-\theta x} \mathcal{W}^{(q)}(x) \mathbf{d}x = (\psi(\theta) - q)^{-1} \mathbf{d}$$

for all *θ* > <sup>Φ</sup>(*q*) = sup {*λ* ≥ 0: *ψ*(*λ*) = *q*}. Recall that

$$
\psi(\theta) = \gamma \theta + \frac{1}{2} \sigma^2 \theta^2 + \int\_0^\infty \left( \mathbf{e}^{-\theta z} - 1 + \theta z \mathbf{1}\_{(0,1]}(z) \right) \nu(\mathbf{d}z),
$$

where *γ* ∈ R and *σ* ≥ 0, and where *ν* is a *σ*-finite measure on (0, <sup>∞</sup>), called the Lévy measure of *X*, satisfying

$$\int\_0^\infty (1 \wedge x^2) \nu(\mathrm{d}x) < \infty.$$

For more details on spectrally negative Lévy processes and scale functions, see e.g., (Kuznetsov et al. 2012; Kyprianou 2014).

In what follows, we will use the following notation: the law of *X* when starting from *X*0 = *x* is denoted by P*x* and the corresponding expectation by E*<sup>x</sup>*. We write P and E when *x* = 0.

Let the spectrally negative Lévy process *X* be the underlying surplus process. A dividend strategy *π* is represented by a non-decreasing, left-continuous and adapted stochastic process *Lπ* = {*L<sup>π</sup>t* , *t* ≥ <sup>0</sup>}, where *<sup>L</sup><sup>π</sup>t* represents the cumulative amount of dividends paid up to time *t* under this strategy, and such that *<sup>L</sup><sup>π</sup>*0 = 0. For a given strategy *π*, the corresponding controlled surplus process *Uπ* = {*U<sup>π</sup>t* , *t* ≥ 0} is defined by *<sup>U</sup><sup>π</sup>t* = *Xt* − *<sup>L</sup><sup>π</sup>t* . The stochastic control problem considered in this paper involves the time of Parisian ruin (with rate *p* > 0) for *Uπ* defined by

$$
\sigma\_p^{\pi^{\pi}} = \inf \left\{ t > 0 \colon t - \mathbf{g}\_t^{\pi} > \mathbf{e}\_p^{\mathcal{S}\_t^{\pi^{\pi}}} \text{ and } \mathcal{U}\_t^{\pi} < 0 \right\},
$$

where *g<sup>π</sup>t* = sup {0 ≤ *s* ≤ *t*: *U<sup>π</sup>s* ≥ <sup>0</sup>}, with **e***g<sup>π</sup><sup>t</sup> p* an independent random variable, following the exponential distribution with mean 1/*p*, associated with the corresponding excursion below 0 (see (Baurdoux et al. 2016) for more details). Please note that, without loss of generality, we have chosen 0 to be the critical level.

**Remark 1.** *Recall that X and Lπ are adapted to the filtration. Set* F∞ = *t*≥0 F*t, i.e., the smallest σ-algebra containing* F*t, for all t* ≥ 0*. It is implicitly assumed that* F∞ *is strictly less than* F *and that all exponential clocks are independent of* F<sup>∞</sup>*.*

A strategy *π* is said to be admissible if a dividend paymen<sup>t</sup> is not larger than the current surplus level, i.e., *<sup>L</sup><sup>π</sup>t*+ − *<sup>L</sup><sup>π</sup>t* ≤ *<sup>U</sup><sup>π</sup>t* , for all *t* < *σπp* , and if no dividends are paid when the controlled surplus is negative, i.e., *t* → *<sup>L</sup><sup>π</sup>t* **<sup>1</sup>**(−∞,<sup>0</sup>)(*U<sup>π</sup><sup>t</sup>* ) ≡ 0. The set of admissible dividend strategies will be denoted by <sup>Π</sup>*p*. These two conditions are motivated by the following interpretation: if *Uπ* enters the interval (−∞, <sup>0</sup>), then a period of *financial distress* begins. Consequently, dividend payments should not cause an excursion under the critical level nor should they be made during those critical periods.

Fix a discounting rate *q* ≥ 0. The value function associated with an admissible dividend strategy *π* ∈ <sup>Π</sup>*p* is defined by

$$v\_{\pi}(\mathbf{x}) = \mathbb{E}\_{\mathbf{x}} \left[ \int\_{0}^{\sigma\_{\text{p}}^{\pi}} \mathbf{e}^{-qt} \mathbf{d}L\_{\text{f}}^{\pi} \right], \quad \mathbf{x} \in \mathbb{R}.$$

The goal is to find the optimal value function *v*∗ defined by

$$v\_\*(\mathbf{x}) = \sup\_{\pi \in \Pi\_p} v\_\pi(\mathbf{x})$$

and an optimal strategy *π*∗ ∈ <sup>Π</sup>*p* such that

$$v\_{\pi\_\*} (\mathfrak{x}) = v\_\* (\mathfrak{x})\_{\prime\prime}$$

for all *x* ∈ R. Because of the Parisian nature of the time of ruin considered in this control problem, we have to deal with possibly negative starting capital.

### *1.2. Main Result and Organization of the Paper*

Let us introduce the family of horizontal barrier strategies, also called reflection strategies. For *b* ∈ R, the (horizontal) barrier strategy at level *b* is the strategy denoted by *π<sup>b</sup>* and with cumulative amount of dividends paid until time *t* given by *Lbt* = sup0<sup>&</sup>lt;*s*≤*<sup>t</sup> Xs* − *b* +, for *t* > 0. If *X*0 = *x* > *b*, then *<sup>L</sup>b*0+ = *x* − *b*. Please note that, if *b* ≥ 0, then *πb* ∈ <sup>Π</sup>*p*. The corresponding value function is thus given by

$$v\_b(\mathbf{x}) = \mathbb{E}\_{\mathbf{x}} \left[ \int\_0^{\sigma\_p^b} \mathbf{e}^{-qt} \mathbf{d}L\_t^b \right] \mathbf{y}$$

for all *x* ∈ R, where *σbp* is the time of Parisian ruin (with rate *p* > 0) for the controlled process *Ubt*= *Xt* − *Lbt*.

Before stating the main result of this paper, recall that the tail of the Lévy measure is the function *x* → *<sup>ν</sup>*(*<sup>x</sup>*, <sup>∞</sup>), where *x* ∈ (0, <sup>∞</sup>), and that a function *f* : (0, ∞) → (0, ∞) is log-convex if the function log(*f*) is convex on (0, <sup>∞</sup>).

**Theorem 1.** *Fix q* ≥ 0 *and p* > 0*. If the tail of the Lévy measure is log-convex, then an optimal strategy for the control problem is formed by a barrier strategy.*

The original version of de Finetti's optimal dividends problem, i.e., when the time of ruin is the first passage time below the critical level (intuitively, when *p* → ∞), has been extensively studied. In a spectrally negative Lévy model, following the work of Avram et al. (2007), an important breakthrough was made by Loeffen (2008); in the latter paper, a sufficient condition, on the Lévy measure *ν*, is given for a barrier strategy to be optimal. This condition was further relaxed by Loeffen and Renaud (2010); in this other paper, it is shown that if the tail of the Lévy measure is log-convex then a barrier strategy is optimal for de Finetti's optimal dividends problem with an affine penalty function at ruin (if we set *S* = *K* = 0 in that paper, we recover the classical problem). To the best of our knowledge, this still stands as the mildest condition for the optimality of a barrier strategy in a spectrally negative Lévy model. Finally, note that Czarna and Palmowski (2014) have considered de Finetti's control problem with deterministic Parisian delays.

The rest of the paper is organized as follows. First, we provide an alternative interpretation of the value function and we fill the gap between the models with classical ruin and no ruin. Then, we compute the value function of an arbitrary horizontal barrier strategy and find the optimal barrier level *b*∗*p* (see the definition in (9)). Finally, we derive the appropriate verification lemma for this control problem and prove that, under our assumption on the Lévy measure, the barrier strategy at level *b*∗*p* is optimal.
