Refraction-reflection strategies in the dual model

TL;DR

This paper analyzes a dual surplus model with capital injection and dividend strategies, establishing the optimality of a refraction-reflection approach and providing numerical insights into dividend optimization.

Contribution

It introduces a novel refraction-reflection strategy for the dual model with capital injection and proves its optimality, extending recent process studies.

Findings

The refraction-reflection strategy is optimal for the dual model with capital injection.

Derived fluctuation identities for the spectrally positive refraction-reflection process.

Numerical results demonstrate the effectiveness of the proposed dividend strategy.

Abstract

We study the dual model with capital injection under the additional condition that the dividend strategy is absolutely continuous. We consider a refraction-reflection strategy that pays dividends at the maximal rate whenever the surplus is above a certain threshold, while capital is injected so that it stays positive. The resulting controlled surplus process becomes the spectrally positive version of the refracted-reflected process recently studied by P\'erez and Yamazaki (2015). We study various fluctuation identities of this process and prove the optimality of the refraction-reflection strategy. Numerical results on the optimal dividend problem are also given.