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On optimal periodic dividend and capital injection strategies for general Lévy models

Dante Mata, Kei Noba, José-Luis Pérez

TL;DR

The paper addresses the bail-out dividend problem under periodic (Poisson) decision times for a surplus modeled by a general Lévy process with two-sided jumps. It develops a pathwise, scale-function–free approach to identify an optimal strategy: a periodic-classical barrier strategy with barrier $a^*$ and continuous capital injections at 0, verified via a variational inequality/HJB framework. The main contributions include a rigorous gradient analysis of the NPV with respect to perturbations, a well-defined candidate barrier determined by a smooth-fit condition, and a complete verification proving optimality. This extends existing results from spectrally one-sided models to general Lévy processes, providing a principled method to implement optimal dividend and capital-injection policies under realistic discrete decision opportunities.

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 Lévy 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.

On optimal periodic dividend and capital injection strategies for general Lévy models

TL;DR

The paper addresses the bail-out dividend problem under periodic (Poisson) decision times for a surplus modeled by a general Lévy process with two-sided jumps. It develops a pathwise, scale-function–free approach to identify an optimal strategy: a periodic-classical barrier strategy with barrier and continuous capital injections at 0, verified via a variational inequality/HJB framework. The main contributions include a rigorous gradient analysis of the NPV with respect to perturbations, a well-defined candidate barrier determined by a smooth-fit condition, and a complete verification proving optimality. This extends existing results from spectrally one-sided models to general Lévy processes, providing a principled method to implement optimal dividend and capital-injection policies under realistic discrete decision opportunities.

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 Lévy 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.
Paper Structure (10 sections, 12 theorems, 78 equations)

This paper contains 10 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lévy processes
  4. The optimal Poissonian dividend problem with classical capital injection
  5. Periodic-classical barrier strategies
  6. Behaviour of $U_a^{(x+\varepsilon)}$ and $U_a^{(x)}$ under $\mathbb{P}_0$
  7. Selection of a candidate optimal threshold
  8. Verification of optimality
  9. The proof of Lemma \ref{['lem_smooth']}
  10. Proof of Lemma \ref{['lemma_HJB2']}

Key Result

Lemma 2.1

If $X$ has unbounded variation paths and the Lévy measure satisfies $\nu(0,\infty) < \infty$ or $\nu(-\infty,0) < \infty$, then Assumption assump2a is satisfied.

Theorems & Definitions (22)

  • Remark 2.1
  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • ...and 12 more