A time of ruin constrained optimal dividend problem for spectrally one-sided L\'evy processes

TL;DR

This paper extends classical optimal dividend problems by incorporating a ruin time constraint within spectrally one-sided Lévy risk models, analyzing both with and without transaction costs, and providing numerical illustrations.

Contribution

It introduces a ruin time constraint into the optimal dividend problem for spectrally one-sided Lévy processes and develops a duality-based solution approach.

Findings

Established no duality gap in the constrained models.

Derived the optimal value function via Lagrangian duality.

Provided numerical examples illustrating the theoretical results.

Abstract

We introduce a longevity feature to the classical optimal dividend problem by adding a constraint on the time of ruin of the firm. We extend the results in \cite{HJ15}, now in context of one-sided L\'evy risk models. We consider de Finetti's problem in both scenarios with and without fix transaction costs, e.g. taxes. We also study the constrained analog to the so called Dual model. To characterize the solution to the aforementioned models we introduce the dual problem and show that the complementary slackness conditions are satisfied and therefore there is no duality gap. As a consequence the optimal value function can be obtained as the pointwise infimum of auxiliary value functions indexed by Lagrange multipliers. Finally, we illustrate our findings with a series of numerical examples.