On the Bail-Out Optimal Dividend Problem

TL;DR

This paper solves the optimal dividend and capital injection problem for spectrally negative Lévy processes with continuous dividend strategies, providing explicit solutions using scale functions and confirming results numerically.

Contribution

It explicitly derives the optimal strategy and value function for the general spectrally negative case using fluctuation identities and scale functions.

Findings

Explicit optimal strategy derived for spectrally negative Lévy processes.

Value function expressed in terms of scale functions.

Numerical results confirm analytical solutions.

Abstract

This paper studies the optimal dividend problem with capital injection under the constraint that the cumulative dividend strategy is absolutely continuous. We consider an open problem of the general spectrally negative case and derive the optimal solution explicitly using the fluctuation identities of the refracted-reflected L\'evy process. The optimal strategy as well as the value function are concisely written in terms of the scale function. Numerical results are also provided to confirm the analytical conclusions.