Optimality of refraction strategies for a constrained dividend problem

Abstract

We consider de Finetti's problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

Optimal Bail-Out Dividend Problem with Transaction Cost and Capital Injection Constraint

“…In this work we are interested in studying the case in which the longevity aspect of the firm is considered, by adding a constraint on expected net present value of injected capital. Similar studies have recently been done in this direction by Hernández et al [8] (see also [9] for the case with absolutely continuous strategies). Following [17], the performance and longevity of the firm remained as two separate problems.…”

Optimality of threshold strategies for spectrally negative Levy processes and a positive terminal value at creeping ruin

“…The optimality of the thresholds strategy when the terminal value is negative is shown and some limited effort on the positive terminal value case has been made. The difficulty investigating the positive terminal value lies in characterizing the appropriate sufficient condition, which shall be the uniqueness of the maximum of θ S , see [7] and [8]. Unfortunately, we did not manage to do this.…”

“…The candidate solution for this type of problem shall be the threshold strategy, see e.g. [7,8,9], which means that any surplus over the threshold level is paid with the maximal admissible dividend rate, while nothing is paid whenever the surplus is under the threshold level. We denote the threshold dividend strategy as π b = (L b t ) t≥0 here.…”

Company Value with Ruin Constraint in Lundberg Models

“…which is defined for s ≥ 0 with ψ(s) ≤ α ≤ 1. Dividend optimization with a ruin related constraint has been considered in several earlier papers: Albrecher and Thonhauser (2007), Hernandez et al (2017) as well as Junca et al (2018) deal with the time value of ruin. In these problems both objective functions are discounted which allows for explicit solutions.…”