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An Optimal Periodic Dividend and Risk Control Problem for an Insurance Company

Mark Kelbert, Harold A. Moreno-Franco

TL;DR

This work solves the optimal control problem of an insurance company choosing periodic shareholder dividends and applying proportional reinsurance, where payout timings are driven by a Poisson process with rate $\gamma$ and risk is managed continuously via reinsurance. Using a diffusion approximation for the surplus, the authors show that the optimal policy, among all periodic-classical strategies, is a barrier-type strategy characterized by two thresholds that depend on $\gamma$ and the reinsurance regime; they derive explicit barrier structures in non-cheap and cheap reinsurance, including very-expensive and expensive subcases, via solutions to nonlinear pseudo-differential systems and a verification theorem. The analysis recovers classical singular-dividend results as $\gamma\to\infty$ and provides detailed constructions of the value functions involving exponentials and gamma-distribution components. Numerically, the barrier strategies converge to the known optimal policies in the continuous-time dividend setting, offering practical guidance on implementing periodic dividend schemes in risk-managed insurance contexts.

Abstract

We study the problem of optimal risk policies and dividend strategies for an insurance company operating under the constraint that the timing of shareholder payouts is governed by the arrival times of a Poisson process. Concurrently, risk control is continuously managed through proportional reinsurance. Our analysis confirms the optimality of a periodic-classical barrier strategy for maximizing the expected net present value until the first instance of bankruptcy across all admissible periodic-classical strategies.

An Optimal Periodic Dividend and Risk Control Problem for an Insurance Company

TL;DR

This work solves the optimal control problem of an insurance company choosing periodic shareholder dividends and applying proportional reinsurance, where payout timings are driven by a Poisson process with rate and risk is managed continuously via reinsurance. Using a diffusion approximation for the surplus, the authors show that the optimal policy, among all periodic-classical strategies, is a barrier-type strategy characterized by two thresholds that depend on and the reinsurance regime; they derive explicit barrier structures in non-cheap and cheap reinsurance, including very-expensive and expensive subcases, via solutions to nonlinear pseudo-differential systems and a verification theorem. The analysis recovers classical singular-dividend results as and provides detailed constructions of the value functions involving exponentials and gamma-distribution components. Numerically, the barrier strategies converge to the known optimal policies in the continuous-time dividend setting, offering practical guidance on implementing periodic dividend schemes in risk-managed insurance contexts.

Abstract

We study the problem of optimal risk policies and dividend strategies for an insurance company operating under the constraint that the timing of shareholder payouts is governed by the arrival times of a Poisson process. Concurrently, risk control is continuously managed through proportional reinsurance. Our analysis confirms the optimality of a periodic-classical barrier strategy for maximizing the expected net present value until the first instance of bankruptcy across all admissible periodic-classical strategies.
Paper Structure (14 sections, 17 theorems, 75 equations, 3 figures, 1 table)

This paper contains 14 sections, 17 theorems, 75 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Choosing an optimal strategy and verification theorem
  4. Verification Theorem
  5. Constructing a solution to the HJB equation and describing the optimal barrier strategy
  6. Non-cheap reinsurance
  7. The very-expensive case
  8. The expensive case
  9. Cheap reinsurance
  10. Numerical results
  11. Proofs of some technical results
  12. Proofs of Propositions \ref{['pr1']} and \ref{['T1']}
  13. Proofs of Propositions \ref{['pr2']}, and \ref{['T2']}
  14. Proofs of Proposition \ref{['pr3']} and \ref{['T3']}

Key Result

Lemma 2.1

If $v\in\mathop{\mathrm{C}}\nolimits^{2}(0,\infty)$ is a concave and increasing function satisfying that $\mathpzc{f}(\cdot;v)$ is decreasing on $(0,\infty)$ and $b^{\gamma}_{1},b^{\gamma}_{2}\in[0,\infty)$, then are achieved at respectively.

Figures (3)

  • Figure 1: Plots of the optimal solution $v_{\gamma,\mathbf{b}^{\gamma}}$ considering \ref{['eq3.1.0']} and \ref{['eq3.6']}, and $v_{\infty}$ with the points $(b^{\gamma}_{2},v_{\gamma,\mathbf{b}^{\gamma}}(b^{\gamma}_{2}))$.
  • Figure 2: Plots of the optimal solution $v_{\gamma,\mathbf{b}^{\gamma}}$ considering \ref{['c6']}, \ref{['c8']} and \ref{['c8.1']} with respect the parameters outlined in the second row of Table \ref{['Ta1']}, and $v_{\infty}$ with the points $(b^{\gamma}_{1},v_{\gamma,\mathbf{b}^{\gamma}}(b^{\gamma}_{1}))$ and $(b^{\gamma}_{2},v_{\gamma,\mathbf{b}^{\gamma}}(b^{\gamma}_{2}))$ indicated by the circles and triangles, respectively.
  • Figure 3: Plots of the optimal solution $v_{\gamma,\mathbf{b}^{\gamma}}$ considering \ref{['c6.2']} and \ref{['c8.0']} with respect the parameters outlined in the third row of Table \ref{['Ta1']}, and $v_{\infty}$ with the points $(b^{\gamma}_{1},v_{\gamma,\mathbf{b}^{\gamma}}(b^{\gamma}_{1}))$ and $(b^{\gamma}_{2},v_{\gamma,\mathbf{b}^{\gamma}}(b^{\gamma}_{2}))$ indicated by the circles and triangles, respectively.

Theorems & Definitions (25)

  • Remark 1
  • Lemma 2.1
  • Remark 2
  • Theorem 2.2: Verification theorem
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Remark 3
  • ...and 15 more