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Optimal Dividend Control with Transaction Costs under Exponential Parisian Ruin for a Refracted Levy Risk Model

Zhongqin Gao, Yan Lv, Jingmin He

TL;DR

The paper studies an impulse dividend control problem for a refracted Lévy risk process with Parisian ruin (exponential delays) and a lower bankruptcy barrier, incorporating transaction costs. It develops a scale-function-based framework: defines the Parisian refracted scale function $\vartheta^{(q+m,q)}$ and derives semi-analytical expressions for exit problems, establishes HJB inequalities, and proves the optimality of a two-barrier impulse strategy $\pi_{(c_1^*,c_2^*)}$ under smoothness conditions. It provides a monotonicity criterion to identify optimal thresholds and proposes a practical numerical scheme. Applications to refracted Brownian and refracted Cramér-Lundberg models with exponential claims demonstrate existence and uniqueness of the optimal impulse strategy and reveal parameter sensitivities.

Abstract

This paper concerns an optimal impulse control problem associated with a refracted Lévy process, involving the reduction of reserves to a predetermined level whenever they exceed a specified threshold. The ruin time is determined by Parisian exponential delays and limited by a lower ultimate bankrupt barrier. We initially obtained the necessary and sufficient conditions for the value function and the optimal impulse control policy. Given a candidate for the optimal strategy, the corresponding expected discounted dividend function is subsequently formulated in terms of the Parisian refracted scale function, which is employed to measure the expected discounted utility of the impulse control. Then, the optimality of the proposed impulse control is verified using the HJB inequalities, and a monotonicity-based criterion is established to identify the admissible region of optimal thresholds, which serves as the basis for the numerical computation of their optimal levels. Finally, we present applications and numerical examples related to Brownian risk process and Cramér-Lundberg process with exponential claims, demonstrating the uniqueness of the optimal impulse strategy and exploring its sensitivity to parameters.

Optimal Dividend Control with Transaction Costs under Exponential Parisian Ruin for a Refracted Levy Risk Model

TL;DR

The paper studies an impulse dividend control problem for a refracted Lévy risk process with Parisian ruin (exponential delays) and a lower bankruptcy barrier, incorporating transaction costs. It develops a scale-function-based framework: defines the Parisian refracted scale function and derives semi-analytical expressions for exit problems, establishes HJB inequalities, and proves the optimality of a two-barrier impulse strategy under smoothness conditions. It provides a monotonicity criterion to identify optimal thresholds and proposes a practical numerical scheme. Applications to refracted Brownian and refracted Cramér-Lundberg models with exponential claims demonstrate existence and uniqueness of the optimal impulse strategy and reveal parameter sensitivities.

Abstract

This paper concerns an optimal impulse control problem associated with a refracted Lévy process, involving the reduction of reserves to a predetermined level whenever they exceed a specified threshold. The ruin time is determined by Parisian exponential delays and limited by a lower ultimate bankrupt barrier. We initially obtained the necessary and sufficient conditions for the value function and the optimal impulse control policy. Given a candidate for the optimal strategy, the corresponding expected discounted dividend function is subsequently formulated in terms of the Parisian refracted scale function, which is employed to measure the expected discounted utility of the impulse control. Then, the optimality of the proposed impulse control is verified using the HJB inequalities, and a monotonicity-based criterion is established to identify the admissible region of optimal thresholds, which serves as the basis for the numerical computation of their optimal levels. Finally, we present applications and numerical examples related to Brownian risk process and Cramér-Lundberg process with exponential claims, demonstrating the uniqueness of the optimal impulse strategy and exploring its sensitivity to parameters.

Paper Structure

This paper contains 18 sections, 19 theorems, 101 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Scale function
  4. Fluctuation identity
  5. Conditions for optimal impulse control
  6. Exit problem
  7. Necessary and sufficient conditions for optimal impulse control
  8. Optimal impulse control strategy
  9. Expected discounted dividend function for impulse strategy $\pi_{(c_1, c_2)}$
  10. Optimality of impulse strategy $\pi_{(c_1^*, c_2^*)}$
  11. Examples
  12. Refracted Brownian risk model
  13. Refracted Cramér-Lundberg model
  14. Proofs and more
  15. Proof of Proposition \ref{['propequexp']}
  16. ...and 3 more sections

Key Result

Proposition 1.1

The function $V_{\pi}$ defined in funcvpix can be equivalently expressed as where $D_{\beta}^{\pi}(t):=\sum_{0\leq s\leq t} \Delta D_s^{\pi}-\beta \mathbf{1}\{\Delta D_s^{\pi}>0\}$ and $L^{\pi}(t) := \int_0^t m \mathbf{1}\{U_s^{\pi} <0\} \mathrm{d} s$.

Figures (6)

  • Figure 1: Impact of parameters on $\vartheta^{(q+m, q)}(x, -l)$ in a refracted Brownian risk model.
  • Figure 2: Impact of parameters on $\vartheta^{(q+m, q)}{'}(x, -l)$ in a refracted Brownian risk model.
  • Figure 3: Impact of parameters on the optimal pair $(c_1^*, c_2^*)$ in a refracted Brownian risk model.
  • Figure 4: Impact of parameters on $\vartheta^{(q+m, q)}(x, -l)$ in a refracted Cramér-Lundberg model.
  • Figure 5: Impact of parameters on $\vartheta^{(q+m, q)}{'}(x, -l)$ in a refracted Cramér-Lundberg model.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Proposition 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 24 more