On the optimality of the refraction--reflection strategy for L\'evy processes

TL;DR

This paper investigates the optimal dividend and capital injection strategy for a general Le9vy process with both positive and negative jumps, proving the optimality of a refraction--reflection strategy.

Contribution

It extends previous work by considering two-sided jumps and establishes the optimality of refraction--reflection strategies in this broader setting.

Findings

Refraction--reflection strategy is proven optimal for the problem.

Existence and uniqueness of solutions for the associated SDEs are established.

The model generalizes previous spectrally one-sided cases.

Abstract

In this paper, we study de Finetti's optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided L\'evy process, however in this paper we use a L\'evy process that may have both positive and negative jumps. In the main theorem, we show that a refraction--reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted L\'evy processes.