Stochastic control with dividend payments and capital injections for Markov additive processes

Kei Noba

arxiv: 2604.00190 · v3 · submitted 2026-03-31 · 🧮 math.PR · math.OC

Stochastic control with dividend payments and capital injections for Markov additive processes

Kei Noba This is my paper

Pith reviewed 2026-05-13 22:35 UTC · model grok-4.3

classification 🧮 math.PR math.OC MSC 60J2593E20

keywords Markov additive processesoptimal dividendscapital injectionsbarrier strategiesstochastic controldynamic programmingregime switching

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The pith

For general Markov additive processes, Markov-modulated periodic-classical barrier strategies are optimal for discrete-time dividend payments with continuous capital injections.

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

The paper examines stochastic control of dividends and capital injections for the additive component of a Markov additive process whose modulating component is a general right process on a Radon space. It derives necessary and sufficient conditions for optimality when dividends are restricted to prescribed discrete times and shows these conditions imply optimality of a class of Markov-modulated periodic-classical barrier strategies. An approximation argument then supplies insight into the form optimal strategies may take when dividends are permitted at arbitrary times. The work matters because it handles models that cannot be reduced to finite-state regime-switching or collections of Lévy problems, relying instead on the additive structure of the process and dynamic programming between dividend opportunities.

Core claim

Necessary and sufficient conditions for optimality of strategies in the discrete-dividend case yield the optimality of Markov-modulated periodic-classical barrier strategies; combining this result with an approximation argument provides insight into possible optimal forms when dividends may occur at arbitrary times, with the proofs developed via the additive structure of MAPs and dynamic programming between dividend opportunities.

What carries the argument

The additive structure of the Markov additive process, which permits dynamic programming between dividend opportunities to characterize optimality for a general right-process modulator without reduction to finite-state Lévy problems.

If this is right

Markov-modulated periodic-classical barrier strategies are optimal for dividend control at discrete times under the derived conditions.
Approximation from the discrete case yields candidates for optimal strategy form when dividends are allowed continuously.
The additive-structure and dynamic-programming argument extends to other stochastic control problems involving general MAPs.

Where Pith is reading between the lines

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

Refining the grid of discrete dividend times could numerically approximate solutions for the continuous-payment case.
Models with diffusion-driven or infinite-state modulators become tractable for dividend optimization without finite-state reduction.
The same structural argument may apply directly to other singular control problems on MAPs where payment times are restricted.

Load-bearing premise

The modulating component is a general right process on a Radon space whose additive structure allows dynamic programming between dividend opportunities to characterize optimality without reduction to finite-state Lévy problems.

What would settle it

A specific Markov additive process with continuous-state modulator for which no Markov-modulated periodic-classical barrier strategy attains the supremum value would falsify the optimality claim.

read the original abstract

Motivated by de Finetti's optimal dividend problem with capital injections, we study a stochastic control problem for the additive component of a Markov additive process (MAP). In contrast to previous studies, the modulating component is allowed to be a general right process on a Radon space, so the model is not restricted to finite-state regime switching and cannot in general be reduced to a finite collection of L\'evy process control problems. Capital injections are allowed at arbitrary times. We first consider the case in which dividend payments are allowed only at prescribed discrete times and establish necessary and sufficient conditions for the optimality of a strategy. These conditions then yield the optimality of a class of Markov-modulated periodic--classical barrier strategies. Combining this optimality result with an approximation argument, we obtain insight into the possible form of optimal strategies in the case where dividend payments, like capital injections, may be made at arbitrary times. Because of the generality of the MAPs considered here, the proof techniques used in previous studies of similar problems are not directly applicable. We therefore develop an alternative argument based on the additive structure of MAPs and dynamic programming between dividend opportunities. The argument also suggests a possible approach to other stochastic control problems involving general MAPs.

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.

Desk Editor's Note private letter to a colleague

This paper derives necessary and sufficient optimality conditions for discrete-time dividend strategies in general MAPs and uses them to characterize Markov-modulated barrier forms, then approximates toward the continuous case.

read the letter

The core advance here is extending de Finetti-type dividend control with capital injections to Markov additive processes whose modulator is a general right process on a Radon space. Prior work stayed with finite-state regime switching that collapses to separate Levy problems; this one keeps the full generality and still gets concrete optimality conditions. The argument relies on the additive structure to run dynamic programming between the discrete dividend times, which produces necessary and sufficient conditions that identify a class of Markov-modulated periodic-classical barrier strategies as optimal. An approximation step then gives some indication of what the continuous-dividend version might look like. That is the actual new content, and the approach is internally consistent on the terms laid out in the abstract. The technique is a reasonable workaround when standard Levy or finite-state methods no longer apply directly. The main soft spots are technical rather than conceptual. The abstract sketches the dynamic-programming setup and the approximation but does not display the full verification, so it is not yet clear how much regularity on the right process is needed to close the arguments or whether the approximation converges in a strong enough sense for the continuous case. These are the usual places where general-state-space control problems can hide gaps, but nothing in the outline suggests a load-bearing flaw or circularity. The paper is aimed at researchers already working on stochastic control for insurance risk or MAPs. A reader who knows the finite-state literature will see the precise extension and the new proof route. It is narrow but cleanly executed. I would send it to peer review; the claims are specific enough that referees can check the derivations, and the generalization is worth having on record.

Referee Report

0 major / 2 minor

Summary. The manuscript studies a stochastic control problem for the additive component of a Markov additive process (MAP) with a general right-process modulating component on a Radon space. Capital injections are permitted at arbitrary times. For the case of dividends paid only at discrete prescribed times, necessary and sufficient optimality conditions are derived; these conditions establish optimality of a class of Markov-modulated periodic-classical barrier strategies. An approximation argument then yields insight into the form of optimal strategies when dividends may be paid continuously. The proofs rely on the additive structure of MAPs and dynamic programming between dividend opportunities rather than reduction to finite-state Lévy problems.

Significance. If the central derivations hold, the work is significant because it treats general MAPs that cannot be reduced to a finite collection of Lévy control problems, thereby extending de Finetti-type dividend problems with capital injections beyond finite-state regime-switching models. The development of an alternative argument based on additive structure and dynamic programming is a clear strength and may apply to other stochastic control problems involving general MAPs. The results supply explicit optimality conditions together with an approximation-based insight into continuous-time strategies.

minor comments (2)

Ensure that the definition and notation for 'Markov-modulated periodic-classical barrier strategies' are introduced with full precision in the main text, including any dependence on the modulating process.
The abstract refers to an 'approximation argument'; verify that the passage from discrete to continuous dividend opportunities is stated with explicit error bounds or convergence statements in the relevant section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for highlighting its potential significance in extending de Finetti-type problems to general Markov additive processes that cannot be reduced to finite-state Lévy models. We appreciate the recognition of our alternative proof approach based on the additive structure and dynamic programming. Below we respond to the assessment.

read point-by-point responses

Referee: If the central derivations hold, the work is significant because it treats general MAPs that cannot be reduced to a finite collection of Lévy control problems... The results supply explicit optimality conditions together with an approximation-based insight into continuous-time strategies.

Authors: We confirm that the central derivations are correct and rely on the additive structure of MAPs together with dynamic programming between dividend opportunities, as detailed in Sections 3 and 4 of the manuscript. The necessary and sufficient optimality conditions for the discrete-time dividend case are established in Theorem 3.1, which directly yields optimality of the Markov-modulated periodic-classical barrier strategies in Theorem 3.2. The approximation argument in Section 5 then provides the indicated insight for the continuous-payment case. We are prepared to supply expanded proof details or clarifications on any specific step if the uncertainty concerns verification of these arguments. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives necessary and sufficient optimality conditions for Markov-modulated periodic-classical barrier strategies using the additive structure of general MAPs and dynamic programming between discrete dividend opportunities. These conditions are obtained directly from the control problem setup without fitting parameters to data or redefining quantities in terms of themselves. The subsequent approximation argument for the continuous-dividend case extends the discrete result independently. No load-bearing self-citations, uniqueness theorems imported from prior work, or ansatzes smuggled via citation are present; the generality of the modulating right process is an explicit modeling choice that enables the DP approach rather than a hidden reduction. The derivation remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of Markov additive processes and right processes; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The modulating component is a general right process on a Radon space
Invoked to allow non-finite-state modulation that cannot be reduced to Levy problems.
domain assumption Additive structure of MAPs permits dynamic programming between dividend opportunities
Central to the alternative argument developed for optimality conditions.

pith-pipeline@v0.9.0 · 5507 in / 1265 out tokens · 30940 ms · 2026-05-13T22:35:01.911077+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We adopt the following procedure to prove optimality: (i) demonstrate particular form of capital injection... (ii) decompose time horizon into intervals between dividend opportunities... use DPP... (iii) characterize conditions using Laplace transform of hitting times.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

Asmussen.Applied probability and queues, Vol

S. Asmussen.Applied probability and queues, Vol. 51 ofApplications of Mathematics (New York). Springer-Verlag, New York, second edition, 2003. Stochastic Modelling and Applied Probability

work page 2003
[2]

Avram, Z

F. Avram, Z. Palmowski, and M. R. Pistorius. On the optimal dividend problem for a spectrally negative L´ evy process.Ann. Appl. Probab., Vol. 17, No. 1, pp. 156–180, 2007

work page 2007
[3]

Bayraktar, A

E. Bayraktar, A. E. Kyprianou, and K. Yamazaki. On optimal dividends in the dual model.Astin Bull., Vol. 43, No. 3, pp. 359–373, 2013

work page 2013
[4]

L. Bo, W. Wang, and K. Yan. De Finetti’s Poissonian dividend control problem under spectrally positive Markov additive process.Stochastic Models, pp. 1–44, 2025

work page 2025
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C ¸ inlar

E. C ¸ inlar. Markov additive processes. I.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 24, pp. 85–93, 1972

work page 1972
[6]

C ¸ inlar

E. C ¸ inlar. Markov additive processes. II.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 24, pp. 95–121, 1972

work page 1972
[7]

J. B. Hiriart-Urruty and C. Lemar´ echal.Fundamentals of convex analysis. Grundlehren Text Editions. Springer-Verlag, Berlin, 2001. Abridged version of ıt Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and ıt II [ibid.; MR1295240 (95m:90002)]

work page 2001
[8]

Jiang and M

Z. Jiang and M. R. Pistorius. Optimal dividend distribution under Markov regime switching.Finance Stoch., Vol. 16, No. 3, pp. 449–476, 2012. 43

work page 2012
[9]

Kuznetsov, A

A. Kuznetsov, A. E. Kyprianou, and J. C. Pardo. Meromorphic L´ evy processes and their fluctuation identities.Ann. Appl. Probab., Vol. 22, No. 3, pp. 1101–1135, 2012

work page 2012
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Kuznetsov, A

A. Kuznetsov, A. E. Kyprianou, and V. Rivero. The theory of scale functions for spectrally negative L´ evy processes. InL´ evy matters II, Vol. 2061 ofLecture Notes in Math., pp. 97–186. Springer, Heidelberg, 2012

work page 2061
[11]

A. E. Kyprianou.Fluctuations of L´ evy processes with applications. Universitext. Springer, Heidelberg, second edition, 2014. Introductory lectures

work page 2014
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A. E. Kyprianou and J. C. Pardo.Stable L´ evy processes via Lamperti-type represen- tations, Vol. 7 ofInstitute of Mathematical Statistics (IMS) Monographs. Cambridge University Press, Cambridge, 2022

work page 2022
[13]

I. B. Lackovi´ c. On the behaviour of sequences of left and right derivatives of a convergent sequence of convex functions.Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 735-762, pp. 19–27, 1982

work page 1982
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R. L. Loeffen. On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative L´ evy processes.Ann. Appl. Probab., Vol. 18, No. 5, pp. 1669– 1680, 2008

work page 2008
[15]

D. Mata, H. A. Moreno-Franco, K. Noba, and J. L. P´ erez. On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes.Nonlinear Anal. Hybrid Syst., Vol. 48, p. Paper No. 101332, 2023

work page 2023
[16]

D. Mata, K. Noba, and J. L. P´ erez. On optimal periodic dividend and capital injection strategies for general L´ evy models, 2025. arXiv:2505.06554

work page arXiv 2025
[17]

Mata L´ opez, K

D. Mata L´ opez, K. Noba, J. L. P´ erez, and K. Yamazaki. Optimal dividends and capital injection: a general L´ evy model with extensions to regime-switching models. Insurance Math. Econom., Vol. 119, pp. 210–225, 2024

work page 2024
[18]

K. Noba. On the optimality of double barrier strategies for L´ evy processes.Stochastic Process. Appl., Vol. 131, pp. 73–102, 2021

work page 2021
[19]

K. Noba. On the optimality of the refraction-reflection strategies for L´ evy processes. Stochastic Process. Appl., Vol. 160, pp. 174–217, 2023

work page 2023
[20]

K. Noba, J. L. P´ erez, K. Yamazaki, and K. Yano. On optimal periodic dividend and capital injection strategies for spectrally negative L´ evy models.J. Appl. Probab., Vol. 55, No. 4, pp. 1272–1286, 2018

work page 2018
[21]

J. L. P´ erez and K. Yamazaki. On the optimality of periodic barrier strategies for a spectrally positive L´ evy process.Insurance Math. Econom., Vol. 77, pp. 1–13, 2017

work page 2017
[22]

Sharpe.General theory of Markov processes, Vol

M. Sharpe.General theory of Markov processes, Vol. 133 ofPure and Applied Math- ematics. Academic Press, Inc., Boston, MA, 1988. 44

work page 1988
[23]

Yamazaki

K. Yamazaki. Inventory control for spectrally positive L´ evy demand processes.Math. Oper. Res., Vol. 42, No. 1, pp. 212–237, 2017

work page 2017
[24]

Yan and W

K. Yan and W. Wang. Optimal dividend and capital injection under Markov mod- ulated spectrally positive risk models.Communications in Statistics - Theory and Methods, Vol. 55, No. 4, pp. 1347–1369, 2026. 45

work page 2026

[1] [1]

Asmussen.Applied probability and queues, Vol

S. Asmussen.Applied probability and queues, Vol. 51 ofApplications of Mathematics (New York). Springer-Verlag, New York, second edition, 2003. Stochastic Modelling and Applied Probability

work page 2003

[2] [2]

Avram, Z

F. Avram, Z. Palmowski, and M. R. Pistorius. On the optimal dividend problem for a spectrally negative L´ evy process.Ann. Appl. Probab., Vol. 17, No. 1, pp. 156–180, 2007

work page 2007

[3] [3]

Bayraktar, A

E. Bayraktar, A. E. Kyprianou, and K. Yamazaki. On optimal dividends in the dual model.Astin Bull., Vol. 43, No. 3, pp. 359–373, 2013

work page 2013

[4] [4]

L. Bo, W. Wang, and K. Yan. De Finetti’s Poissonian dividend control problem under spectrally positive Markov additive process.Stochastic Models, pp. 1–44, 2025

work page 2025

[5] [5]

C ¸ inlar

E. C ¸ inlar. Markov additive processes. I.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 24, pp. 85–93, 1972

work page 1972

[6] [6]

C ¸ inlar

E. C ¸ inlar. Markov additive processes. II.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 24, pp. 95–121, 1972

work page 1972

[7] [7]

J. B. Hiriart-Urruty and C. Lemar´ echal.Fundamentals of convex analysis. Grundlehren Text Editions. Springer-Verlag, Berlin, 2001. Abridged version of ıt Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and ıt II [ibid.; MR1295240 (95m:90002)]

work page 2001

[8] [8]

Jiang and M

Z. Jiang and M. R. Pistorius. Optimal dividend distribution under Markov regime switching.Finance Stoch., Vol. 16, No. 3, pp. 449–476, 2012. 43

work page 2012

[9] [9]

Kuznetsov, A

A. Kuznetsov, A. E. Kyprianou, and J. C. Pardo. Meromorphic L´ evy processes and their fluctuation identities.Ann. Appl. Probab., Vol. 22, No. 3, pp. 1101–1135, 2012

work page 2012

[10] [10]

Kuznetsov, A

A. Kuznetsov, A. E. Kyprianou, and V. Rivero. The theory of scale functions for spectrally negative L´ evy processes. InL´ evy matters II, Vol. 2061 ofLecture Notes in Math., pp. 97–186. Springer, Heidelberg, 2012

work page 2061

[11] [11]

A. E. Kyprianou.Fluctuations of L´ evy processes with applications. Universitext. Springer, Heidelberg, second edition, 2014. Introductory lectures

work page 2014

[12] [12]

A. E. Kyprianou and J. C. Pardo.Stable L´ evy processes via Lamperti-type represen- tations, Vol. 7 ofInstitute of Mathematical Statistics (IMS) Monographs. Cambridge University Press, Cambridge, 2022

work page 2022

[13] [13]

I. B. Lackovi´ c. On the behaviour of sequences of left and right derivatives of a convergent sequence of convex functions.Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 735-762, pp. 19–27, 1982

work page 1982

[14] [14]

R. L. Loeffen. On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative L´ evy processes.Ann. Appl. Probab., Vol. 18, No. 5, pp. 1669– 1680, 2008

work page 2008

[15] [15]

D. Mata, H. A. Moreno-Franco, K. Noba, and J. L. P´ erez. On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes.Nonlinear Anal. Hybrid Syst., Vol. 48, p. Paper No. 101332, 2023

work page 2023

[16] [16]

D. Mata, K. Noba, and J. L. P´ erez. On optimal periodic dividend and capital injection strategies for general L´ evy models, 2025. arXiv:2505.06554

work page arXiv 2025

[17] [17]

Mata L´ opez, K

D. Mata L´ opez, K. Noba, J. L. P´ erez, and K. Yamazaki. Optimal dividends and capital injection: a general L´ evy model with extensions to regime-switching models. Insurance Math. Econom., Vol. 119, pp. 210–225, 2024

work page 2024

[18] [18]

K. Noba. On the optimality of double barrier strategies for L´ evy processes.Stochastic Process. Appl., Vol. 131, pp. 73–102, 2021

work page 2021

[19] [19]

K. Noba. On the optimality of the refraction-reflection strategies for L´ evy processes. Stochastic Process. Appl., Vol. 160, pp. 174–217, 2023

work page 2023

[20] [20]

K. Noba, J. L. P´ erez, K. Yamazaki, and K. Yano. On optimal periodic dividend and capital injection strategies for spectrally negative L´ evy models.J. Appl. Probab., Vol. 55, No. 4, pp. 1272–1286, 2018

work page 2018

[21] [21]

J. L. P´ erez and K. Yamazaki. On the optimality of periodic barrier strategies for a spectrally positive L´ evy process.Insurance Math. Econom., Vol. 77, pp. 1–13, 2017

work page 2017

[22] [22]

Sharpe.General theory of Markov processes, Vol

M. Sharpe.General theory of Markov processes, Vol. 133 ofPure and Applied Math- ematics. Academic Press, Inc., Boston, MA, 1988. 44

work page 1988

[23] [23]

Yamazaki

K. Yamazaki. Inventory control for spectrally positive L´ evy demand processes.Math. Oper. Res., Vol. 42, No. 1, pp. 212–237, 2017

work page 2017

[24] [24]

Yan and W

K. Yan and W. Wang. Optimal dividend and capital injection under Markov mod- ulated spectrally positive risk models.Communications in Statistics - Theory and Methods, Vol. 55, No. 4, pp. 1347–1369, 2026. 45

work page 2026