*3.3. Optimal Barrier Level*

As defined in (Loeffen 2008; Loeffen and Renaud 2010), the optimal barrier level in de Finetti's classical control problem is given by

$$b^\*\_{\infty} = \sup \left\{ b \ge 0 \colon \mathcal{W}^{(q)\prime}(b) \le \mathcal{W}^{(q)\prime}(\mathfrak{x}), \text{ for all } \mathfrak{x} \ge 0 \right\}.$$

Similarly, let us define the candidate for the optimal barrier level for the current version of this control problem by

$$b\_p^\* = \sup \left\{ b \ge 0 \colon Z\_q'(b, \Phi(p+q)) \le Z\_q'(\mathbf{x}, \Phi(p+q)), \text{ for all } \mathbf{x} \ge 0 \right\}.\tag{9}$$

**Proposition 2.** *Fix q* ≥ 0 *and p* > 0*. Suppose the tail of the Lévy measure is log-convex. Then,* 0 ≤ *b*∗ *p* ≤ *b*∗ ∞ *and b*∗ *p* > 0 *if and only if*

$$
\Phi(p+q) - p\mathcal{W}^{(q)}(0) < \frac{p}{\Phi(p+q)} \mathcal{W}^{(q)\prime}(0+). \tag{10}
$$

*Equivalently, b*∗ *p* > 0 *if and only if one of the following three cases hold:*


$$\frac{c\Phi(p+q)}{p}\left(\Phi(p+q) - \frac{p}{c}\right) < \frac{q + \nu(0, \infty)}{c}r$$

$$where \ c = \gamma + \int\_0^1 \mathfrak{x}\nu(\mathsf{dx}).$$

### **Proof.** See the proof in Appendix B.

First of all, note from Proposition 2 that the optimal barrier level *b*∗ *p*, when Parisian ruin with rate *p* is implemented, is always lower than the optimal barrier level *b*∗ ∞ when classical ruin is used.

In cases (a) and (c), the value of *b*∗ *p* can be either positive or zero, depending on the parameters of the model. It is clear from the condition in (10) that, when *q* > 0, if the Parisian rate *p* is small enough (large delays), then *b*∗ *p* = 0; in words, if Parisian delays are infinite (no ruin), then it is better to start paying out dividends right away. However, when *q* = 0 (no discounting), if Parisian delays are infinite (no ruin), then *b*∗ *p* > 0 if and only if <sup>E</sup>[*<sup>X</sup>*1] > 0.

Also, in case (a), if *Xt* = *ct* + *σBt* is a Brownian motion with drift, then

$$
\Phi(p+q) = \frac{1}{\sigma^2} \left( \sqrt{c^2 + 2\sigma^2(p+q)} - c \right)
$$

and we can verify that *b*∗*p* = 0 as soon as the Brownian coefficient *σ* is large enough.

**Remark 3.** *In Section 4 of (Avram and Minca 2017), economic principles for evaluating the efficiency of a surplus process are discussed. One of them is that the optimal barrier level be equal to zero.*

Interestingly, the condition in (c) can be re-written as follows:

$$\frac{c\Phi(p+q)}{p} \to \left[ \int\_0^{\sigma\_\infty^0} \mathbf{e}^{-qt} \mathbf{d}L\_t^0 \right] < \mathbb{E} \left[ \int\_0^{\sigma\_p^0} \mathbf{e}^{-qt} \mathbf{d}L\_t^0 \right] = \upsilon\_0(0).$$

Indeed, when *σ* = 0 and *ν*(0, ∞) < <sup>∞</sup>, it is known (see Equation (3.14) in (Avram et al. 2007)) that

$$\mathbb{E}\left[\int\_0^{\sigma\_{\infty}^0} \mathbf{e}^{-qt} \mathbf{d}L\_t^0 \right] = \frac{c}{q + \nu(0, \infty)}$$

and, from Proposition 1, we have

$$\mathbb{E}\left[\int\_0^{\sigma\_p^0} \mathbf{e}^{-qt} \mathbf{d}L\_t^0 \right] = \frac{1}{\Phi(p+q) - pW^{(q)}(0+)}.$$

### **4. Verification Lemma and Proof of the Main Result**

Define the operator Γ associated with *X* by

$$
\Gamma \upsilon(\mathbf{x}) = \gamma \upsilon'(\mathbf{x}) + \frac{\sigma^2}{2} \upsilon''(\mathbf{x}) + \int\_0^\infty \left( \upsilon(\mathbf{x} - \mathbf{z}) - \upsilon(\mathbf{x}) + \upsilon'(\mathbf{x}) z \mathbf{1}\_{(0,1]}(\mathbf{z}) \right) \nu(\mathbf{dz}), \tag{11}
$$

where *v* is a function defined on R such that <sup>Γ</sup>*v*(*x*) is well defined. We say that a function *v* is sufficiently smooth if it is continuously differentiable on (0, ∞) when *X* is of bounded variation and twice continuously differentiable on (0, ∞) when *X* is of unbounded variation.

Next is the verification lemma of our stochastic control problem. As the controlled process is now allowed to spend time below the critical level, it is different from the classical verification lemma (see (Loeffen 2008)).

**Lemma 1.** *Let* Γ *be the operator defined in* (11)*. Suppose that π*ˆ ∈ <sup>Π</sup>*p is such that vπ*<sup>ˆ</sup> *is sufficiently smooth and that, for all x* ∈ R*,*

$$\left(\Gamma - q - p\mathbf{1}\_{\left(-\infty,0\right)}\right) v\_{\mathbb{H}}(x) \le 0$$

*and, for all x* > 0*, <sup>v</sup>π*<sup>ˆ</sup>(*x*) ≥ 1*. In this case, π*ˆ *is an optimal strategy for the control problem.* **Proof.** Set *w* := *vπ*<sup>ˆ</sup> and let *π* ∈ <sup>Π</sup>*p* be an arbitrary admissible strategy. As *w* is sufficiently smooth, applying an appropriate change-of-variable/version of Ito's formula to the joint process *t*, *t*0 **<sup>1</sup>**(−∞,<sup>0</sup>)(*U<sup>π</sup><sup>r</sup>* )d*<sup>r</sup>*, *<sup>U</sup><sup>π</sup>t* yields

$$\begin{cases} & \mathbf{e}^{-qt-p\int\_{0}^{t} \mathbf{1}\_{(-\infty,0)}(\mathsf{U}^{\pi}\_{r})d\mathbf{r}} w\left(\mathcal{U}^{\pi}\_{t}\right) - w\left(\mathcal{U}^{\pi}\_{0}\right) \\ & = \int\_{0}^{t} \mathbf{e}^{-qs-p\int\_{0}^{s} \mathbf{1}\_{(-\infty,0)}(\mathsf{U}^{\pi}\_{r})d\mathbf{r}} \left[ \left(\mathbf{I}-q\right)w\left(\mathcal{U}^{\pi}\_{s}\right) - p\mathbf{1}\_{(-\infty,0)}\left(\mathcal{U}^{\pi}\_{s}\right)w\left(\mathcal{U}^{\pi}\_{s}\right) \right] \mathbf{d}s \\ & - \int\_{0}^{t} \mathbf{e}^{-qs-p\int\_{0}^{s} \mathbf{1}\_{(-\infty,0)}(\mathsf{U}^{\pi}\_{r})d\mathbf{r}} w'\left(\mathcal{U}^{\pi}\_{s-}\right) \mathbf{d}L\_{s}^{\pi} + M\_{t}^{\pi} \\ & + \sum\_{0 < s \le t} \mathbf{e}^{-qs-p\int\_{0}^{s} \mathbf{1}\_{(-\infty,0)}(\mathsf{U}^{\pi}\_{r})d\mathbf{r}} \left[ w\left(\mathcal{U}^{\pi}\_{s-} - \Delta L\_{s}^{\pi}\right) - w\left(\mathcal{U}^{\pi}\_{s-}\right) + w'\left(\mathcal{U}^{\pi}\_{s-}\right)\Delta L\_{s}^{\pi} \right], \end{cases} \tag{12}$$

where *Mπ* = {*M<sup>π</sup>t* , *t* ≥ 0} is a (local) martingale.

Consider an independent (of F∞) Poisson process with intensity measure *p* d*t* and jump times *Tpi*, *i* ≥ 1. Therefore, we can write

$$\mathbf{e}^{-p\int\_{0}^{t}\mathbf{1}\_{\left(-\alpha\beta\right)}\left(\mathbf{l}\mathbf{l}^{\pi}\right)\mathbf{d}r} = \mathbb{P}\_{\mathbf{x}}\left(T\_{l}^{p} \notin \{r \in \left(0,\mathbf{s}\right] \colon \mathbf{l}\mathbf{l}\_{r}^{\pi} < 0\}, \text{ for all } i \ge 1 \,\middle|\,\mathcal{F}\_{\infty}\right) = \mathbb{E}\_{\mathbf{x}}\left[\mathbf{1}\_{\left\{\sigma\_{p}^{\pi} > s\right\}} \,|\,\mathcal{F}\_{\infty}\right]$$

and consequently

$$\mathbb{E}\_{\mathbf{x}}\left[\int\_{0}^{t}\mathbf{e}^{-qs-p}\,\big|\,^{t}\mathbf{1}\_{(-\infty,0)}(\mathcal{U}^{\pi}\_{r})\mathrm{d}\mathbf{r}\,\big|\,^{\pi}\mathrm{d}\mathbf{L}\_{\mathbf{s}}^{\pi}\right] = \mathbb{E}\_{\mathbf{x}}\left[\int\_{0}^{t}\mathbf{e}^{-qs}\,\mathbb{E}\_{\mathbf{x}}\left[\mathbf{1}\_{\left\{\sigma^{\pi}\_{p}>s\right\}}\big|\,\mathcal{F}\_{\infty}\right]\mathrm{d}\mathbf{L}\_{\mathbf{s}}^{\pi}\right] \\ = \mathbb{E}\_{\mathbf{x}}\left[\int\_{0}^{\sigma^{\pi}\_{p}\wedge t}\mathrm{e}^{-qs}\mathrm{d}\mathbf{L}\_{\mathbf{s}}^{\pi}\right],$$

where we used the definition of a Riemann-Stieltjes integral and the monotone convergence theorem for conditional expectations.

Now, as for all *x* ∈ R,

$$\left(\Gamma - q - p\mathbf{1}\_{\left(-\infty,0\right)}\right)w(\mathfrak{x}) \le 0$$

and, for all *x* > 0, *w*(*x*) ≥ 1, using standard arguments (see e.g., (Loeffen 2008)) and our definition of an admissible strategy, e.g., that *Lπ* is identically zero when *Uπ* is below zero, we ge<sup>t</sup>

$$\mathbb{E}w(\mathbf{x}) \ge \mathbb{E}\_{\mathbf{x}} \left[ \int\_0^\infty \mathbf{e}^{-qs-p} \int\_0^s \mathbf{1}\_{(-\infty,0)}(\mathcal{U}^\pi\_r) \mathbf{d} \mathcal{U}^\pi\_s \right] \\ = \mathbb{E}\_{\mathbf{x}} \left[ \int\_0^{\sigma^\pi\_p} \mathbf{e}^{-qs} \mathbf{d} \mathcal{U}^\pi\_s \right] = v\_\pi(\mathbf{x}).$$

This concludes the proof.

The rest of this section is devoted to proving Theorem 1, i.e., proving that an optimal strategy for the control problem is formed by the barrier strategy at level *b*∗ := *b*∗*p*.

By the definition of *b*∗ given in (9), for 0 ≤ *x* ≤ *b*<sup>∗</sup>, we have

$$w'\_{b^\*} (x) = \frac{Z'\_q(x, \Phi(p+q))}{Z'\_q(b^\*, \Phi(p+q))} \ge 1.$$

By the definition of *vb*∗ , for *x* > *b*<sup>∗</sup>, we have *vb*∗ (*x*) = 1. This means *vb*∗ (*x*) ≥ 1, for all *x* ≥ 0. Please note that for any *x* ∈ R, we have

$$\begin{array}{rcl} (\Gamma - q - p) \mathbf{e}^{\Phi(p+q)\mathbf{x}} &=& \mathbf{e}^{\Phi(p+q)\mathbf{x}} \left( \gamma \Phi(p+q) + \frac{\sigma^{2}}{2} \Phi^{2}(p+q) \right) \\ &+ \mathbf{e}^{\Phi(p+q)\mathbf{x}} \left[ \int\_{0}^{\infty} \left( \mathbf{e}^{-\Phi(p+q)\mathbf{z}} - 1 + \Phi(p+q)z \mathbf{1}\_{(0,1]}(\mathbf{z}) \right) \nu(\mathbf{d}\mathbf{z}) - (q+p) \right] \\ &=& \mathbf{e}^{\Phi(p+q)\mathbf{x}} \left[ \Psi \left( \Phi(p+q) \right) - (q+p) \right] = 0. \end{array} \tag{13}$$

Consequently, for *x* < 0, we have

$$(\Gamma - q - p) \, Z\_q(\mathfrak{x}\_\prime \Phi(p + q)) = 0$$

and, for *x* ≥ 0, using (6), we have

$$(\Gamma - q)\, Z\_q(\mathbf{x}, \Phi(p+q)) = p \int\_0^\infty \mathbf{e}^{\Phi(p+q)y} \left(\Gamma - q\right) W^{(q)}(\mathbf{x} + y) d\mathbf{y} = 0, \mathbf{y}$$

since (Γ − *q*) *<sup>W</sup>*(*q*)(*x*) = 0 for all *x* > 0 (see e.g., (Biffis and Kyprianou 2010)). Please note that under our assumption, *W*(*q*) is sufficiently smooth. Indeed, by Theorem 1.2 in (Loeffen and Renaud 2010), if the tail of the Lévy measure is log-convex, then *W*(*q*) is log-convex. Therefore, *<sup>W</sup>*(*q*)(*x*) exists and is continuous for almost all *x* ∈ (0, ∞); see e.g., (Roberts and Varberg 1973).

As a consequence, and since *vb*∗ is smooth in *x* = *b*<sup>∗</sup>, we have

$$\left(\Gamma - q - p\mathbf{1}\_{\left(-\infty,0\right)}\right)v\_{b^\*}\left(\mathbf{x}\right) = 0, \quad \text{for } \mathbf{x} \le b^\*.$$

All that is now left to verify is that (Γ − *q*) *vb*∗ (*x*) ≤ 0, for all *x* > *b*<sup>∗</sup>. It can be done following the same steps as in the proof of Theorem 2 in (Loeffen 2008), thanks to the fact that, under our assumption on the Lévy measure, the function *<sup>Z</sup>q*(·, <sup>Φ</sup>(*p* + *q*)) is sufficiently smooth (see the details in Appendix B). The details are left to the reader.

**Funding:** This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC).

**Acknowledgments:** Let me thank two anonymous referees for their diligence and comments which improved this final version of the paper. Special thanks to Ronnie Loeffen, for providing the interpretation in Remark 2 and other comments which improved the final presentation, and to Florin Avram, for pointing out important literature which had been overlooked in a previous version. With their comments and suggestions, they both contributed to Section 2.

**Conflicts of Interest:** The author declares no conflict of interest.
