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De Finetti's Poissonian Dividend Control Problem under Spectrally Positive Markov Additive Process

Lijun Bo, Wenyuan Wang, Kaixin Yan

TL;DR

The paper addresses De Finetti's optimal dividend problem with capital injections in a regime-switching surplus model driven by a spectrally positive Markov additive process. It develops a Poisson observation framework where dividends are paid at Poisson epochs and injections prevent bankruptcy, and solves the problem by first analyzing an auxiliary problem with a terminal payoff on a single SPLP, obtaining explicit double-barrier strategies via scale-function and excursion-based techniques. A fixed-point dynamic-programming approach then extends the auxiliary solution to the original MAP setting, proving the existence of a regime-modulated double barrier Poissonian-continuous-reflection strategy that is optimal. Numerical experiments illustrate the optimal barriers and how model parameters affect policy design, highlighting the practical applicability of the regime-switching double-barrier framework in stochastic dividend control.

Abstract

We study a De Finetti's optimal dividend and capital injection problem under a Markov additive model. The surplus process without dividend and capital injection is assumed to follow a spectrally positive Markov additive process (MAP). Dividend payments are made at the jump times of an independent Poisson process and capitals are injected to avoid bankruptcy. The aim of the paper is to characterize an optimal dividend and capital injection strategy that maximizes the expected total discounted dividends subtracted by the total discounted costs of capital injection. Applying the fluctuation and excursion theory for Levy processes and the stochastic control theory, we first address an auxiliary dividend and capital injection control problem with a terminal payoff under the spectrally positive Levy model. Using results obtained for this auxiliary problem and a fixed point argument for iterations induced by dynamic program, we characterize the optimal strategy of our prime control problem as a regime-modulated double-barrier Poissonian-continuous-reflection dividend and capital injection strategy. Besides, a numerical example is provided to illustrate the features of the optimal strategies. The impacts of model parameters are also studied.

De Finetti's Poissonian Dividend Control Problem under Spectrally Positive Markov Additive Process

TL;DR

The paper addresses De Finetti's optimal dividend problem with capital injections in a regime-switching surplus model driven by a spectrally positive Markov additive process. It develops a Poisson observation framework where dividends are paid at Poisson epochs and injections prevent bankruptcy, and solves the problem by first analyzing an auxiliary problem with a terminal payoff on a single SPLP, obtaining explicit double-barrier strategies via scale-function and excursion-based techniques. A fixed-point dynamic-programming approach then extends the auxiliary solution to the original MAP setting, proving the existence of a regime-modulated double barrier Poissonian-continuous-reflection strategy that is optimal. Numerical experiments illustrate the optimal barriers and how model parameters affect policy design, highlighting the practical applicability of the regime-switching double-barrier framework in stochastic dividend control.

Abstract

We study a De Finetti's optimal dividend and capital injection problem under a Markov additive model. The surplus process without dividend and capital injection is assumed to follow a spectrally positive Markov additive process (MAP). Dividend payments are made at the jump times of an independent Poisson process and capitals are injected to avoid bankruptcy. The aim of the paper is to characterize an optimal dividend and capital injection strategy that maximizes the expected total discounted dividends subtracted by the total discounted costs of capital injection. Applying the fluctuation and excursion theory for Levy processes and the stochastic control theory, we first address an auxiliary dividend and capital injection control problem with a terminal payoff under the spectrally positive Levy model. Using results obtained for this auxiliary problem and a fixed point argument for iterations induced by dynamic program, we characterize the optimal strategy of our prime control problem as a regime-modulated double-barrier Poissonian-continuous-reflection dividend and capital injection strategy. Besides, a numerical example is provided to illustrate the features of the optimal strategies. The impacts of model parameters are also studied.
Paper Structure (8 sections, 22 theorems, 122 equations, 2 figures, 6 tables)

This paper contains 8 sections, 22 theorems, 122 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation and Main Result
  3. Auxiliary Singular Control Problem
  4. Proof of Theorem \ref{['thm2.1']}
  5. Numerical Illustration
  6. Preliminaries of Spectrally Positive Lévy Processes
  7. Proof of Proposition \ref{['V.x']}
  8. Proofs of Lemma \ref{['lem.pm']}

Key Result

Theorem 2.1

There exists a vector $\overrightarrow{b^{*}}\in\mathbb{R}_{+}^{I}$ such that $(D_t^{0,\overrightarrow{b^{*}}},R_t^{0,\overrightarrow{b^{*}}})_{t\geq 0}$ (which is defined by eq:DbRb with $\overrightarrow{b}$ replaced as $\overrightarrow{b^{*}}$) is an optimal strategy for the prime problem eq:value where $V_{0,\overrightarrow{b^{*}}}(x,i)$ is given by eq:valuefcn... but with $\overrightarrow{b}$

Figures (2)

  • Figure 1: Iteration of functions $J_{\widehat{V}_n}(x,1;\pi^{b^{\widehat{V}_n}_1})$ and $J_{\widehat{V}_n}(x,2;\pi^{b^{\widehat{V}_n}_2})$ for $1\leq n \leq 61$.
  • Figure 2: The uncontrolled surplus process $X_t$ and the controlled surplus process $U_t$ with the dividend barriers $\overrightarrow{b^*}=(3.81,3.97)$.

Theorems & Definitions (37)

  • Theorem 2.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof : Proof of Proposition \ref{['LTkappa0-']}
  • Lemma 3.6
  • proof
  • ...and 27 more