# On the optimality of double barrier strategies for L\'evy processes

## TL;DR

This paper proves the optimality of double barrier strategies for dividend problems in Le9vy risk models with both positive and negative jumps, extending previous spectrally one-sided results.

## Contribution

It generalizes existing theorems to Le9vy processes with two-sided jumps, demonstrating the optimality of double barrier strategies beyond spectrally one-sided cases.

## Key findings

- Double barrier strategies are optimal for Le9vy processes with two-sided jumps.
- Scale functions are insufficient for analyzing these strategies in the two-sided case.
- Sample path analysis is used to establish properties of expected NPVs.

## Abstract

This paper studies de Finetti's optimal dividend problem with capital injection. We confirm the optimality of a double barrier strategy when the underlying risk model follows a L\'evy process that may have positive and negative jumps. The main result in this paper is a generalization of Theorem 3 in Avram et al.(2007), which is the spectrally negative case, and Theorem 3.1 in Bayraktar et al.(2013), which is the spectrally positive case. In contrast with the spectrally one-sided cases, double barrier strategies cannot be handled by using scale functions to obtain some properties of the expected net present values (NPVs) of dividends and capital injections. Instead, to obtain these properties, we observe changes in the sample path (and the associated NPV) when there is a slight change to the initial value or the barrier value.