*2.1. De Finetti's Problem*

De Finetti's problem corresponds to the case where *k* = <sup>∞</sup>, implying that *C* ≡ 0, that is, capital injections cannot be profitable. In this case, the controlled process is ruined as soon as it goes below zero. For this problem, the optimal value function will be denoted by *VdF*.

It is well-known that for this problem, constant barrier strategies are very important. For *b* ≥ 0, the (horizontal) barrier strategy at level *b* is the strategy with a cumulative amount of dividends paid until time *t* > 0 given by *Dbt* = sup0<sup>&</sup>lt;*s*≤*<sup>t</sup> Xs* − *b* +. If *X*0 = *x* > *b*, then *<sup>D</sup>b*0+ = *x* − *b* (a lump sum paymen<sup>t</sup> is made). For such a strategy, the value function is such that *J*(*<sup>x</sup>*, 0, *D<sup>b</sup>*) = *<sup>V</sup><sup>b</sup>*(*x*), where

$$V^b(\mathbf{x}) := \mathbb{E}\_x \left[ \int\_0^{\tau^b} \mathbf{e}^{-qt} \mathbf{d} D\_t^b \right] \mathbf{x}$$

where *τb* is the time of ruin for the controlled process *Ubt* = *Xt* − *Dbt* . In this case, P*x τb* < ∞ = 1. It

 is well-known that, for a SNLP (see, for example, Avram et al. 2007),

$$V^b(\mathbf{x}) = \begin{cases} \frac{W\_q(\mathbf{x})}{W\_q'(b)}, & \mathbf{x} \le b, \\ \mathbf{x} - b + \frac{W\_q(b)}{W\_q(b)}, & \mathbf{x} > b, \end{cases} \tag{6}$$

where the *q*-scale function *Wq* (Bertoin 1998) is given through its Laplace transform:

$$\int\_0^\infty \mathbf{e}^{-\theta x} \mathcal{W}\_\theta(x) \mathbf{dx} = \frac{1}{\psi(\theta) - q'} \tag{7}$$

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

It is known (see Theorem 1.1 in (Loeffen and Renaud 2010)) that if the tail of the jump distribution is log-convex, then an optimal dividend policy is formed by the barrier strategy at level *b*<sup>∗</sup>, where *b*∗ is the last maximum of the *barrier function*

$$H^{dF}(b) = \frac{1}{\mathcal{W}'\_q(b)}, \ b > 0. \tag{8}$$

In this case, the optimal value function *VdF* is given by *Vb*<sup>∗</sup>.

Consequently, for a Cramér–Lundberg risk process with exponentially distributed claims, the optimal value function *VdF* is equal to the value function of a barrier strategy. More precisely, for *X* given in (1) with exponential jumps, we have

$$\mathcal{W}\_{\mathfrak{q}}(\mathfrak{x}) = \frac{A\_{+}\mathbf{e}^{\rho\_{+}\chi} - A\_{-}\mathbf{e}^{\rho\_{-}\chi}}{c(\rho\_{+} - \rho\_{-})}, \mathfrak{x} \ge 0,$$

where

$$\rho\_{\pm} = \frac{1}{2c} \left( -\left( \mu c - \lambda - q \right) \pm \sqrt{\left( \mu c - \lambda - q \right)^{2} + 4\mu qc} \right)$$

are such that *ρ*− ≤ 0 ≤ *ρ*+ = <sup>Φ</sup>(*q*), and where *A*± = *μ* + *ρ*<sup>±</sup>. In this case, the barrier function *HdF* has a unique maximum at level

$$b^\* = \begin{cases} \frac{1}{\rho\_+ - \rho\_-} \log \left( \frac{\rho\_-^2 \left( \mu + \rho\_- \right)}{\rho\_+^2 \left( \mu + \rho\_+ \right)} \right) & \text{if } (q + \lambda)^2 - c\lambda\mu < 0, \\ 0 & \text{if } (q + \lambda)^2 - c\lambda\mu \ge 0, \end{cases}$$

and we have *VdF* = *Vb*<sup>∗</sup>.
