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arxiv: 1907.04162 · v1 · pith:Y7P3KVYAnew · submitted 2019-07-09 · 🧮 math.PR

Optimality of impulse control problem in refracted L\'evy model with Parisian ruin and transaction costs

Irmina Czarna , Adam Kaszubowski This is my paper

Pith reviewed 2026-05-25 00:08 UTC · model grok-4.3

classification 🧮 math.PR
keywords impulse controloptimal dividendsrefracted Lévy processParisian ruintransaction costsq-scale functionsBrownian motionCramér-Lundberg model
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The pith

In refracted Lévy models with Parisian ruin, a unique (c1,c2) impulse policy is optimal for the dividend problem with transaction costs.

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

The paper derives explicit formulas for the Parisian refracted q-scale functions in the linear Brownian motion case and the Cramér-Lundberg model with exponential claims. It then uses these formulas to establish sufficient conditions under which an impulse policy that reduces the surplus to level c1 whenever it exceeds c2 is optimal, and to prove that this optimal policy is unique within the class. A sympathetic reader would care because transaction costs rule out continuous intervention, so identifying the precise intervention thresholds determines the maximum expected discounted dividends before ruin occurs under delayed ruin detection.

Core claim

Using newly derived analytical formulas for the Parisian refracted q-scale functions, the paper shows that for the linear Brownian motion and the Cramér-Lundberg process with exponential claims there exists a unique (c1,c2) policy which is optimal for the impulse control problem.

What carries the argument

The (c1,c2) impulse policy, which reduces reserves to a fixed lower level c1 whenever they exceed an upper level c2, together with the Parisian refracted q-scale functions that give the expected discounted dividends and ruin probabilities under the controlled refracted dynamics.

If this is right

  • Sufficient conditions are given under which any (c1,c2) policy is optimal.
  • For Brownian motion and the exponential-claims model the optimal policy is unique.
  • Closed-form expressions for the Parisian refracted q-scale functions become available in these two cases.
  • Numerical illustrations confirm that the thresholds can be computed explicitly from the scale-function formulas.

Where Pith is reading between the lines

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

  • The same verification technique could be tried on other Lévy processes once analogous scale functions are obtained.
  • The explicit thresholds supply a practical benchmark for testing more general numerical optimization routines.
  • Insurance companies facing both transaction costs and delayed ruin detection could use the (c1,c2) rule as a simple implementable dividend rule.

Load-bearing premise

The optimum is attained inside the restricted class of (c1,c2) policies and the new scale-function formulas correctly compute the performance measures under refraction and Parisian delay.

What would settle it

An explicit calculation, for one of the two models, showing that the value function obtained from the candidate (c1,c2) thresholds is strictly less than the value achieved by some other admissible impulse strategy, or that the derived scale functions fail to solve the governing integro-differential equations.

Figures

Figures reproduced from arXiv: 1907.04162 by Adam Kaszubowski, Irmina Czarna.

Figure 1
Figure 1. Figure 1: Plot of the V (q) function for linear Brownian motion Note that the shape of this function is similar as for the classic scale function for linear Brownian motion. In the next picture we will consider V (q)0 with the optimal points (c ∗ 1 , c∗ 2 ) and β = 0.05 [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the V (q)0 function for the linear Brownian motion and the optimal pair (c ∗ 1 , c∗ 2 ). Transaction cost equal to 0.05 The first interesting observation is the shape of this function for x < 0. One can see that (c ∗ 1 , c∗ 2 ) belongs to the set B from the Proposition 3. One can also observe that our optimal pair (c ∗ 1 , c∗ 2 ) satisfy condition from the Theorem 10. Moreover, one can be intereste… view at source ↗
Figure 3
Figure 3. Figure 3: Plot of the V (q)0 function for the linear Brownian motion and optimal pair (c ∗ 1 , c∗ 2 ) not belonging to the set B. Case for β = 1 Thus, one can see that depending on the parameters of the process one can get different possibilities from the Proposition 3. 4.2. Cram´er-Lundberg process. In the second example we will consider the Cram´er-Lundberg process Xt = pt − X Nt i=1 Ui , where p > 0, {Ui} ∞ i=1 i… view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the V (q) for Cram´er-Lundberg As in the linear Brownian motion setting, one can also see the similar shape of Parisian scale function with the shape of classical scale function. However, even if this is not directly clear from the [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the V (q)0 for the Cram´er-Lundberg process and the optimal pair (c ∗ 1 , c∗ 2 ). Transaction cost is equal to 0.02 One can see from the [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plot of the V (q)0 for the Cram´er-Lundberg process and the optimal pair (c ∗ 1 , c∗ 2 ) not belonging to the set B. Case for β = 1 One can see from the [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
read the original abstract

In this paper we investigate an optimal dividend problem with transaction costs, where the surplus process is modelled by a refracted L\'evy process and the ruin time is considered with Parisian delay. Presence of the transaction costs implies that one need to consider the impulse control problem as a control strategy in such model. An impulse policy $(c_1,c_2)$, which is to reduce the reserves to some fixed level $c_1$ whenever they are above another level $c_2$ is an important strategy for the impulse control problem. Therefore, we give sufficient conditions under which the above described impulse policy is optimal. Further, we give the new analytical formulas for the Parisian refracted $q$-scale functions in the case of the linear Brownian motion and the Cr\'amer-Lundberg process with exponential claims. Using these formulas we show that for these models there exists a unique $(c_1, c_2)$ policy which is optimal for the impulse control problem. Numerical examples are also provided.

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

0 major / 3 minor

Summary. The paper studies an optimal dividend problem with transaction costs under a refracted Lévy surplus process with Parisian ruin. It states general sufficient conditions for optimality of (c1,c2) impulse policies, derives explicit Parisian refracted q-scale functions for linear Brownian motion and the Cramér-Lundberg model with exponential claims by solving the associated integro-differential equations, substitutes these into the performance criterion to identify the unique pair (c1,c2) satisfying the conditions, and verifies optimality by checking the quasi-variational inequalities, with numerical examples.

Significance. If the derivations hold, the explicit scale-function formulas and the self-contained optimality verification for the two concrete models constitute a useful contribution to the literature on impulse control of risk processes. The work supplies closed-form expressions rather than numerical approximations and reduces optimality to direct verification of the QVI at the identified thresholds.

minor comments (3)
  1. The sufficient conditions for optimality of the (c1,c2) policy are stated in the abstract and introduction but would benefit from being collected and numbered as a formal theorem immediately before the scale-function derivations.
  2. In the sections deriving the Parisian refracted q-scale functions for the two models, the boundary conditions used to solve the integro-differential equations should be listed explicitly with equation numbers to facilitate checking the subsequent substitution step.
  3. The numerical examples section would be strengthened by reporting the exact parameter values (drift, volatility, claim rate, etc.) used to generate the plots of the value function and the optimal thresholds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the paper and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations self-contained

full rationale

The paper derives new explicit formulas for Parisian refracted q-scale functions by solving the associated integro-differential equations for the refracted dynamics (linear Brownian motion and Cramér-Lundberg with exponential claims). These formulas are then substituted into the performance criterion to identify the unique (c1,c2) thresholds satisfying the paper's stated sufficient conditions for optimality, which are verified by direct checking of the quasi-variational inequalities. No step reduces a prediction or optimality claim to a fitted input by construction, nor relies on load-bearing self-citation of unverified uniqueness results. The argument is independent of external fitted quantities and remains within the model's defining equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the modeling choice of a refracted Lévy process with Parisian ruin and on the existence of the associated scale functions that are used to verify optimality.

axioms (2)
  • domain assumption Surplus dynamics follow a refracted Lévy process
    Core modeling assumption stated in title and abstract.
  • domain assumption Ruin is measured with a fixed Parisian delay
    Explicitly invoked to define the ruin time in the control problem.

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