Optimality of refraction strategies for a constrained dividend problem

TL;DR

This paper analyzes optimal refraction strategies for a constrained dividend problem in spectrally one-sided Lévy risk models, establishing threshold strategies as optimal and characterizing solutions via duality and numerical examples.

Contribution

It introduces a dual Lagrangian approach to characterize optimal strategies under ruin constraints in spectrally one-sided Lévy models.

Findings

Threshold strategies are optimal for the dividend problem.

The dual Lagrangian approach effectively characterizes the optimal solutions.

Numerical examples illustrate the theoretical results.

Abstract

We consider de Finetti's problem for spectrally one-sided L\'evy 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.