On optimal periodic dividend and capital injection strategies for general L\'evy models

TL;DR

This paper studies an optimal dividend and capital injection strategy for general Lévy processes, showing that a periodic reflection strategy maximizes expected dividends while maintaining non-negative surplus.

Contribution

It introduces and proves the optimality of a periodic-classical reflection strategy for Lévy models with jumps, extending classical dividend problem solutions.

Findings

Optimal strategy is a periodic-classical reflection at a fixed level.

Strategy ensures surplus remains non-negative with minimal injections.

Results apply to general Lévy processes with positive and negative jumps.

Abstract

We consider a version of de Finetti's dividend problem, with the bail-out contraint to keep the surplus non-negative, and where dividend payments can only be made at the arrival times of an independent Poisson process. For a general L\'evy process with positive and negative jumps, we show the optimality of a periodic-classical reflection strategy that pays the excess above a given level at each Poisson arrival time, and also reflects below at 0 in the classical sense.