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arxiv: 2409.12523 · v2 · pith:K7KBU4MGnew · submitted 2024-09-19 · 🧮 math.OC · math.PR

A dynamic optimal reinsurance strategy with capital injections in the Cramer-Lundberg model

Zakaria Aljaberi , Asma Khedher , Mohamed Mnif This is my paper

Pith reviewed 2026-05-23 20:39 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords Cramer-Lundberg modeloptimal reinsuranceHamilton-Jacobi-Bellman equationdividend barriercapital injectionsruin timeproportional reinsurancestochastic control
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The pith

The value function for optimal reinsurance and dividends in the Cramer-Lundberg model solves the Hamilton-Jacobi-Bellman equation to give explicit structure equations for the proportional reinsurance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an explicit form for the optimal dynamic reinsurance strategy combined with dividend payments in a classical risk model that permits capital injections. The goal is to maximize the expected value of discounted dividends paid out until the company is ruined. By treating the problem as a stochastic control problem and solving the associated Hamilton-Jacobi-Bellman equation, the authors characterize the optimal policy completely in terms of two barriers: one for stopping large overshoots and one for dividend payments. A reader would care because this turns an abstract optimization into a set of solvable equations that insurance firms could use to set their reinsurance levels and payout rules.

Core claim

In the Cramer-Lundberg framework with proportional reinsurance and capital injections, the optimal strategy maximizes expected discounted dividends until ruin by stopping at the first overshoot below zero that exceeds limit a and paying dividends at upper barrier b. The value function is identified as a particular solution to the Hamilton-Jacobi-Bellman equation, which yields an exhaustive explicit characterization of the optimal policy through comprehensive structure equations for the proportional reinsurance.

What carries the argument

the Hamilton-Jacobi-Bellman equation for the value function of the reinsurance and dividend control problem, solved to produce structure equations for the reinsurance proportion

If this is right

  • The optimal policy stops at the first time the overshoot below zero exceeds limit a and pays dividends when the reserve reaches upper barrier b.
  • The reinsurance proportion is determined explicitly by the structure equations obtained from the HJB solution.
  • The method applies to proportional reinsurance treaties as illustrated by the examples in the paper.
  • Capital injections are allowed to keep the surplus process running until the stopping time defined by the overshoot rule.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit structure equations could be solved numerically to find the best values of barriers a and b for any given claim distribution.
  • The same HJB approach might be applied to risk models with claim arrival processes other than Poisson.
  • The barriers derived here could be compared against simulation-based optimization to test sensitivity to the claim size distribution.

Load-bearing premise

The admissible policies are restricted to the specific form of stopping at a fixed overshoot threshold a and paying dividends at a fixed barrier b, with the HJB verification holding for the standard Cramer-Lundberg dynamics without extra regularity conditions on the claim distribution.

What would settle it

For an exponential claim size distribution, derive the structure equations from the HJB solution, obtain the explicit value function and barriers, then run Monte Carlo simulations of the controlled surplus process and check whether the simulated expected discounted dividends match the analytical value.

Figures

Figures reproduced from arXiv: 2409.12523 by Asma Khedher, Mohamed Mnif, Zakaria Aljaberi.

Figure 1
Figure 1. Figure 1: The optimal reinsurance strategy α ∗ We notice in [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The optimal value function [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The optimal value function (value function at x=0) [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The optimal value function and reinsurance [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: surplus process with and without reinsurance strategy [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The surplus process with Reinsurance (changes every 1 a [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The surplus process with Reinsurance (changes every 1, [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The optimal reinsurance strategy α ∗ (changes every time unit) [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The optimal reinsurance strategy α ∗ (changes every 12 time units) 28 [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
read the original abstract

In this article we consider the surplus process of an insurance company within the Cramer-Lundberg framework. We study the optimal reinsurance strategy and dividend distribution of an insurance company under proportional reinsurance, in which capital injections are allowed. Our aim is to find a general dynamic reinsurance strategy that maximises the expected discounted cumulative dividends until the time of passage below a given level, called ruin. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a, and only pay dividends when the reserve reaches an upper barrier b. Using analytical methods, we identify the value function as a particular solution to the associated Hamilton Jacobi Bellman equation. This approach leads to an exhaustive and explicit characterisation of optimal policy. The proportional reinsurance is given via comprehensive structure equations. Furthermore we give some examples illustrating the applicability of this method for proportional reinsurance treaties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies optimal dynamic proportional reinsurance and dividend strategies in the classical Cramer-Lundberg risk model when capital injections are permitted. The objective is to maximize expected discounted cumulative dividends until ruin. Admissible controls are restricted a priori to policies that terminate upon an overshoot below zero exceeding a threshold a and pay dividends only when the surplus reaches an upper barrier b. The authors assert that analytical solution of the associated Hamilton-Jacobi-Bellman equation yields an explicit characterization of the value function and the optimal reinsurance proportion via structure equations, with illustrative examples provided for specific treaties.

Significance. If the derivation and verification hold, the work supplies an explicit analytical solution to a combined reinsurance-dividend control problem that extends standard Cramer-Lundberg models by incorporating capital injections and a specific policy structure. The structure equations could enable direct computation and comparison across treaties, representing a concrete advance in insurance risk management literature when the classical HJB solution is rigorously justified.

major comments (2)
  1. [HJB derivation and verification (abstract and main analytical sections)] The central claim of an exhaustive explicit characterization rests on the candidate value function satisfying the integro-differential HJB equation pointwise in the classical sense and on a verification theorem that directly yields optimality. For arbitrary claim distributions (no density or bounded variation assumed), the generator is not guaranteed to act classically; the manuscript does not state the required regularity conditions or invoke viscosity solutions. This is load-bearing for the explicit-policy conclusion.
  2. [Model formulation and admissible policies] The admissible set is restricted at the outset to the (a,b)-form with capital injections. No separate argument establishes that an optimal policy must lie in this class independently of the HJB solution; optimality is shown only within the restricted class. This assumption underpins the claim of an exhaustive characterization of the optimal policy.
minor comments (2)
  1. [Introduction and model setup] Notation for the overshoot threshold a and dividend barrier b should be introduced with explicit definitions and distinguished from the ruin level in the first section where they appear.
  2. [Examples] The examples section would benefit from a brief statement of the specific claim-size distributions used and the numerical values chosen for a and b to facilitate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report on our manuscript. We address the major comments point by point below, with proposed revisions to improve clarity and rigor where the concerns are valid.

read point-by-point responses

  1. Referee: [HJB derivation and verification (abstract and main analytical sections)] The central claim of an exhaustive explicit characterization rests on the candidate value function satisfying the integro-differential HJB equation pointwise in the classical sense and on a verification theorem that directly yields optimality. For arbitrary claim distributions (no density or bounded variation assumed), the generator is not guaranteed to act classically; the manuscript does not state the required regularity conditions or invoke viscosity solutions. This is load-bearing for the explicit-policy conclusion.

    Authors: We agree that explicit regularity conditions for classical satisfaction of the HJB equation should be stated. The derivation in the manuscript proceeds under the assumption that the value function is twice differentiable, which is valid when the claim size distribution admits a density (as in the illustrative examples). In the revision we will add a dedicated remark specifying these conditions and clarifying that the explicit structure equations and verification hold in the classical sense under them. For fully general distributions without density we note that viscosity solutions provide an alternative framework, but this does not affect the analytical results for the cases considered. revision: yes

  2. Referee: [Model formulation and admissible policies] The admissible set is restricted at the outset to the (a,b)-form with capital injections. No separate argument establishes that an optimal policy must lie in this class independently of the HJB solution; optimality is shown only within the restricted class. This assumption underpins the claim of an exhaustive characterization of the optimal policy.

    Authors: The restriction to (a,b)-policies is introduced at the model-formulation stage because barrier-type strategies with capital injections are the natural candidates suggested by the problem structure and by known optimality results in related dividend problems without reinsurance. Optimality is established within this class via the HJB solution and verification. We acknowledge that a separate proof that no superior policy exists outside the class would require additional arguments (e.g., via direct comparison or dynamic programming principles). In the revision we will explicitly qualify the main claims to state that the characterization is exhaustive within the considered admissible class, thereby removing any ambiguity. revision: partial

Circularity Check

0 steps flagged

No significant circularity; standard HJB solution for controlled risk process

full rationale

The derivation applies the classical Hamilton-Jacobi-Bellman optimality principle to the Cramer-Lundberg surplus process under proportional reinsurance and capital injections. The value function is identified as the solution to the associated integro-differential equation, yielding explicit barrier-type policies. This is the standard dynamic-programming reduction in stochastic control and does not reduce any claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. No load-bearing step equates an output to its input by construction, and the approach remains self-contained against the model primitives.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Cramer-Lundberg assumptions and the HJB verification principle; no free parameters or invented entities are identifiable from the abstract.

axioms (2)
  • domain assumption The risk process is a Cramer-Lundberg process with Poisson claim arrivals and general claim size distribution under proportional reinsurance.
    This is the modeling framework stated in the abstract.
  • domain assumption The value function satisfies the Hamilton-Jacobi-Bellman equation with appropriate boundary conditions at the barriers a and b.
    Invoked to obtain the explicit characterization of the optimal policy.

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