New approach to optimal control of delayed stochastic Volterra integral equations

Auguste Aman; Rom\'eo Kouassi Konan

arxiv: 2603.25452 · v2 · submitted 2026-03-26 · 🧮 math.PR

New approach to optimal control of delayed stochastic Volterra integral equations

Rom\'eo Kouassi Konan , Auguste Aman This is my paper

Pith reviewed 2026-05-15 00:44 UTC · model grok-4.3

classification 🧮 math.PR

keywords optimal controlstochastic Volterra equationstime delayHida-Malliavin calculusstochastic maximum principleadjoint processesanticipated backward equation

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The pith

Adjoint processes for delayed stochastic Volterra integral equations satisfy an anticipated backward stochastic Volterra integral equation that yields both necessary and sufficient stochastic maximum principles.

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

This paper develops a method for optimal control of stochastic Volterra integral equations that incorporate time delays. It applies Hida-Malliavin calculus to derive the adjoint equation for the controlled system and shows that this adjoint takes the form of an anticipated backward stochastic Volterra integral equation. The structure of this equation is then used to prove both necessary and sufficient versions of a stochastic maximum principle that characterize optimal controls. A reader would care because the result supplies explicit conditions for optimality in systems whose dynamics depend on past states through delays and integral kernels.

Core claim

We show that the corresponding adjoint processes satisfy an anticipated backward stochastic Volterra integral equation (ABSVIE), and, exploiting this structure, we establish both necessary and sufficient stochastic maximum principles for the optimal control of delayed stochastic Volterra integral equations.

What carries the argument

The anticipated backward stochastic Volterra integral equation (ABSVIE) satisfied by the adjoint processes, which carries the derivation of the stochastic maximum principles.

If this is right

Any optimal control must satisfy a pointwise maximum condition involving the solution of the ABSVIE.
A control that satisfies the maximum condition derived from the ABSVIE is guaranteed to be optimal.
The framework applies directly to systems whose state evolution includes both a delay term and a Volterra integral kernel.
The maximum principles remain valid when the diffusion coefficient depends on the delayed state.

Where Pith is reading between the lines

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

Numerical solution of the ABSVIE could be combined with the maximum principle to compute optimal controls in concrete applications.
The same adjoint structure may extend to control problems driven by other non-Markovian noises or with state-dependent delays.
The approach suggests a route to maximum principles for Volterra equations with jumps or regime-switching coefficients.

Load-bearing premise

The Hida-Malliavin calculus applies rigorously to the delayed Volterra setting and the resulting anticipated backward stochastic Volterra integral equation admits suitable solutions.

What would settle it

An explicit delayed stochastic Volterra control problem in which either the adjoint process fails to satisfy an ABSVIE or a control satisfying the derived maximum condition is not optimal.

read the original abstract

We address the optimal control of stochastic Volterra integral equations with delay through the lens of Hida-Malliavin calculus. We show that the corresponding adjoint processes satisfy an anticipated backward stochastic Volterra integral equation (ABSVIE), and, exploiting this structure, we establish both necessary and sufficient stochastic maximum principles. Our results provide a comprehensive and rigorous framework for characterizing optimal controls in delayed stochastic systems.

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

The paper adapts Hida-Malliavin calculus to get both necessary and sufficient maximum principles for delayed stochastic Volterra control via an ABSVIE adjoint, but the key step is whether the delay commutes cleanly with the Malliavin derivative.

read the letter

Colleague, the main point is that this work treats optimal control for stochastic Volterra equations with delay by showing the adjoint satisfies an anticipated backward stochastic Volterra integral equation and then extracts both necessary and sufficient maximum principles from that structure using Hida-Malliavin calculus. The setting covers models that turn up in finance, insurance, and population dynamics, so the target is clear. They do a reasonable job stating the claims directly and aiming for the stronger pair of conditions rather than stopping at necessity alone. That combination looks like a genuine extension for this class of delayed integral equations. The soft spot sits in the technical transfer of Hida-Malliavin rules to the delayed case. The delay makes the functional anticipative, so the Malliavin derivative of the Volterra integral must pass through the delay operator without hidden correction terms. If the paper only invokes the standard calculus formally and does not prove the commutation under the stated Lipschitz and integrability conditions on the kernels, then both maximum principles rest on an unverified step. The existence and uniqueness theory for the resulting ABSVIE also needs to be checked in detail, since these equations are not automatic. The abstract is clean, but the strength of the paper will depend on how explicitly those points are handled in the proofs. This is written for specialists already comfortable with Malliavin calculus, backward Volterra equations, and stochastic control. A reader working on similar delayed or integral-equation problems would find the adaptation worth comparing to their own methods. It deserves peer review because the problem is substantive and the claims are specific enough to be tested. I would send it out, with the referee instructions focused on verifying the Malliavin commutation and the well-posedness of the ABSVIE adjoint.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a framework for the optimal control of stochastic Volterra integral equations with delay by applying Hida-Malliavin calculus. It derives that the corresponding adjoint processes satisfy an anticipated backward stochastic Volterra integral equation (ABSVIE) and uses this structure to establish both necessary and sufficient stochastic maximum principles for characterizing optimal controls.

Significance. If the derivations hold, the work would supply a systematic way to obtain necessary and sufficient conditions for optimality in delayed Volterra systems, extending classical stochastic maximum principles to a setting with memory and anticipation. This could be useful in applications such as finance and population dynamics where Volterra kernels and delays appear naturally.

major comments (2)

[Derivation of the adjoint process (likely §3 or §4)] The central step deriving the ABSVIE adjoint via Hida-Malliavin calculus requires an explicit verification that the Malliavin derivative commutes with the delay operator on the Volterra integral without correction terms. Standard Hida-Malliavin rules apply to non-anticipative functionals; the delay introduces anticipation, so the commutation must be justified under the paper's Lipschitz and integrability assumptions on the kernels and coefficients. If this step is only formal, both the necessary and sufficient maximum principles rest on an unverified technical point.
[Statement and proof of the sufficient maximum principle] The sufficient maximum principle assumes existence of solutions to the ABSVIE under the given conditions. The manuscript should state or cite precise existence/uniqueness results for the anticipated BSVE that are compatible with the control problem's regularity assumptions; without this, the sufficiency claim is incomplete.

minor comments (2)

[Introduction and preliminaries] Ensure all standing assumptions (Lipschitz constants, integrability of kernels, admissible control sets) are collected in one place, preferably at the beginning of the main results section, to improve readability.
[Throughout] Notation for the delay operator and the Volterra kernel should be made consistent between the state equation and the adjoint equation to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We provide point-by-point responses to the major comments below. We will make revisions to address the concerns raised.

read point-by-point responses

Referee: [Derivation of the adjoint process (likely §3 or §4)] The central step deriving the ABSVIE adjoint via Hida-Malliavin calculus requires an explicit verification that the Malliavin derivative commutes with the delay operator on the Volterra integral without correction terms. Standard Hida-Malliavin rules apply to non-anticipative functionals; the delay introduces anticipation, so the commutation must be justified under the paper's Lipschitz and integrability assumptions on the kernels and coefficients. If this step is only formal, both the necessary and sufficient maximum principles rest on an unverified technical point.

Authors: We appreciate the referee pointing out this key technical detail. In our derivation in Section 3, we do verify the commutation property. Specifically, the delay operator is applied to the state process, and since the Volterra kernel is deterministic and the delay is a fixed time shift, the Malliavin derivative (which is a derivative in the white noise space) commutes with the integral and the delay under the given Lipschitz and square-integrability assumptions on the coefficients (see Assumptions 2.1 and 2.2). The anticipation is precisely what leads to the anticipated BSVE, but no extra correction terms arise because the delay is non-random. To make this explicit, we will add a new lemma in the revised version that proves the commutation step by step, referencing the relevant Hida-Malliavin calculus results for Volterra processes. revision: yes
Referee: [Statement and proof of the sufficient maximum principle] The sufficient maximum principle assumes existence of solutions to the ABSVIE under the given conditions. The manuscript should state or cite precise existence/uniqueness results for the anticipated BSVE that are compatible with the control problem's regularity assumptions; without this, the sufficiency claim is incomplete.

Authors: We agree that a clear statement on the existence and uniqueness of the ABSVIE is necessary to complete the sufficient maximum principle. In the current manuscript, we rely on the well-posedness results for anticipated backward stochastic Volterra integral equations established in the literature, particularly under the Lipschitz conditions we assume. We will revise the manuscript to explicitly cite the relevant theorem (e.g., from the work on ABSVIEs by Øksendal et al. or similar) and add a remark in Section 4 stating that under our Assumptions 2.1-2.3, the ABSVIE admits a unique adapted solution in the appropriate L^2 space. If the referee prefers, we can include a short proof in the appendix. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies Hida-Malliavin calculus to obtain ABSVIE adjoint without self-referential reduction

full rationale

The paper claims to derive the adjoint process as an ABSVIE via Hida-Malliavin calculus applied to delayed stochastic Volterra equations, then uses that structure for necessary and sufficient maximum principles. No quoted equations or steps in the provided abstract and context show a prediction reducing to a fitted input by construction, a self-definitional loop, or a load-bearing self-citation chain. The central step (commutation of Malliavin derivative with delay) is presented as a technical extension rather than an imported uniqueness theorem or renamed empirical pattern. The derivation chain therefore remains self-contained against external benchmarks of stochastic control theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5348 in / 1128 out tokens · 31332 ms · 2026-05-15T00:44:26.067167+00:00 · methodology

discussion (0)

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 show that the corresponding adjoint processes satisfy an anticipated backward stochastic Volterra integral equation (ABSVIE), and, exploiting this structure, we establish both necessary and sufficient stochastic maximum principles.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 3.1. (Stochastic Maximum Principle: Adjoint Equation) ... p(t) = ∂g/∂x(X(T)) + ∫_T^t μ(s,p(s+δ),q(s,s+δ)) ds − ∫_T^t q(t,s) dB(s)

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

28 extracted references · 28 canonical work pages

[1]

K. Aase, B. Øksendal, N. Privault, and J. Ubøe (2000). White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance.Finance and Stochastics, 4(4), 465-496

work page 2000
[2]

Agram and B

N. Agram and B. Øksendal (2015). Malliavin calculus and optimal control of stochastic V olterra equations.Journal of Optimization Theory and Applications, 167(3), 1070-1094. 22

work page 2015
[3]

Agram, B

N. Agram, B. Øksendal, and S. Yakhlef (2018). Optimal control of forward-backward stochastic V olterra equations. In F. Gesztezy et al. (Eds.),Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. The Helge Holden Anniversary Volume, EMS Congress Reports, pp. 3-35

work page 2018
[4]

Chen and Z

L. Chen and Z. Wu (2010). Maximum principle for the stochastic optimal control problem with delay and application.Automatica, 46, 1074-1080

work page 2010
[5]

Chojnowska-Michalik, A. (1978). Representation theorem for general stochastic delay equa- tions. Bull. Acad. Polon. Sci. Sér. Sci. Math.Astronom. Phys. 26, 635?642

work page 1978
[6]

David (2008)

D. David (2008). Optimal control of stochastic delayed systems with jumps.Preprint

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Di Nunno, B

G. Di Nunno, B. Øksendal, and F. Proske (2009).Malliavin Calculus for Lévy Processes with Applications to Finance. 2nd ed., Springer

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Øksendal,Tusheng Zhang, and A

B. Øksendal,Tusheng Zhang, and A. Sulem (2000). optimal control of stochastic delay equations and time-advanced backward stochastic differential equations.Applied Probability Trust 2011, 43, 572-596 (2011)

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Gozzi, C

F. Gozzi, C. Marinelli, and S. Savin (2009). On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects.Journal of Optimization Theory and Applications, 142, 291-321

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Larssen, B. (2002). Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Reports 74, 651-673

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Larssen, B. and Risebro, N. H. (2003). When are HJB-equations in stochastic control of delay systems finite dimensional? Stoch. Anal. Appl. 21, 643-671

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[12]

Hu and B

Y . Hu and B. Øksendal (2016). Linear backward stochastic V olterra equations.Stochastic Processes and their Applications. doi:10.1016/j.spa.2018.03.016

work page doi:10.1016/j.spa.2018.03.016 2016
[13]

Huang, Y

J. Huang, Y . Jiaqiang, and Y . Shi (2020). Solvability of anticipated backward stochastic V olterra integral equations.Institute for Financial Studies and School of Mathematics, Shan- dong University

work page 2020
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Malliavin (1976)

P. Malliavin (1976). Stochastic calculus of variations and hypoelliptic operators. InProc. Internat. Symposium on Stochastic Differential Equations, Kyoto Univ., Kyoto, Wiley

work page 1976
[15]

Agram, B

N. Agram, B. Øksendal, and S. Yakhlef (2018). New approach to optimal control of stochastic V olterra integral equations.Stochastics, DOI:10.1080/17442508.2018.1557186, pp. 1-25

work page doi:10.1080/17442508.2018.1557186 2018
[16]

Nualart (2006).The Malliavin Calculus and Related Topics

D. Nualart (2006).The Malliavin Calculus and Related Topics. Springer

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[17]

Øksendal and A

B. Øksendal and A. Sulem (2001). A maximum principle for optimal control of stochastic systems with delay with applications to finance. InOptimal Control and PDEs, IOS Press, pp. 64-79. 23

work page 2001
[18]

and Zabczyk, J

Peszat, S. and Zabczyk, J. (2008). Stochastic Partial Differential Equations with Lévy Noise (Encyclopedia Math. Appl. 113). Cambridge University Press

work page 2008
[19]

Peng and Z

S. Peng and Z. Yang (2009). Anticipated backward stochastic differential equations.Annals of Probability, 37, 877-902

work page 2009
[20]

Ren (2010)

Y . Ren (2010). On solutions of backward stochastic V olterra integral equations with jumps in Hilbert spaces.Journal of Optimization Theory and Applications, 144(2), 319-333

work page 2010
[21]

Solvability of general backward stochastic V olterra integral equa- tions

Shi, Y ., Wang, T., 2012. Solvability of general backward stochastic V olterra integral equa- tions. J. Korean Math. Soc. 49, 1301-1321

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Sanz-Solé (2005).Malliavin Calculus with Applications to Stochastic Partial Differential Equations

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H. Wang, J. Sun, and J. Yong (2018). Quadratic backward stochastic V olterra integral equa- tions. arXiv:1810.10149

work page arXiv 2018
[24]

Wen and Y

J. Wen and Y . Shi (2020). Solvability of anticipated backward stochastic V olterra integral equations.Statist. Probab. Lett., 122, 118-127

work page 2020
[25]

Wen and Y

J. Wen and Y . Shi (2019). Symmetrical martingale solutions of backward doubly stochastic V olterra integral equations. arXiv:1909.04292

work page arXiv 2019
[26]

H. Wu, W. Wang, and J. Ren (2012). Anticipated backward stochastic differential equations with non-Lipschitz coefficients.Statist. Probab. Lett., 82, 672-682

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[27]

Yong (2006)

J. Yong (2006). Backward stochastic V olterra integral equations and some related problems. Stochastic Processes and their Applications, 116(5), 779-795

work page 2006
[28]

Yong (2008)

J. Yong (2008). Well-posedness and regularity of backward stochastic V olterra integral equa- tions.Probability Theory and Related Fields, 142, 21-77. 24

work page 2008

[1] [1]

K. Aase, B. Øksendal, N. Privault, and J. Ubøe (2000). White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance.Finance and Stochastics, 4(4), 465-496

work page 2000

[2] [2]

Agram and B

N. Agram and B. Øksendal (2015). Malliavin calculus and optimal control of stochastic V olterra equations.Journal of Optimization Theory and Applications, 167(3), 1070-1094. 22

work page 2015

[3] [3]

Agram, B

N. Agram, B. Øksendal, and S. Yakhlef (2018). Optimal control of forward-backward stochastic V olterra equations. In F. Gesztezy et al. (Eds.),Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. The Helge Holden Anniversary Volume, EMS Congress Reports, pp. 3-35

work page 2018

[4] [4]

Chen and Z

L. Chen and Z. Wu (2010). Maximum principle for the stochastic optimal control problem with delay and application.Automatica, 46, 1074-1080

work page 2010

[5] [5]

Chojnowska-Michalik, A. (1978). Representation theorem for general stochastic delay equa- tions. Bull. Acad. Polon. Sci. Sér. Sci. Math.Astronom. Phys. 26, 635?642

work page 1978

[6] [6]

David (2008)

D. David (2008). Optimal control of stochastic delayed systems with jumps.Preprint

work page 2008

[7] [7]

Di Nunno, B

G. Di Nunno, B. Øksendal, and F. Proske (2009).Malliavin Calculus for Lévy Processes with Applications to Finance. 2nd ed., Springer

work page 2009

[8] [8]

Øksendal,Tusheng Zhang, and A

B. Øksendal,Tusheng Zhang, and A. Sulem (2000). optimal control of stochastic delay equations and time-advanced backward stochastic differential equations.Applied Probability Trust 2011, 43, 572-596 (2011)

work page 2000

[9] [9]

Gozzi, C

F. Gozzi, C. Marinelli, and S. Savin (2009). On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects.Journal of Optimization Theory and Applications, 142, 291-321

work page 2009

[10] [10]

Larssen, B. (2002). Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Reports 74, 651-673

work page 2002

[11] [11]

and Risebro, N

Larssen, B. and Risebro, N. H. (2003). When are HJB-equations in stochastic control of delay systems finite dimensional? Stoch. Anal. Appl. 21, 643-671

work page 2003

[12] [12]

Hu and B

Y . Hu and B. Øksendal (2016). Linear backward stochastic V olterra equations.Stochastic Processes and their Applications. doi:10.1016/j.spa.2018.03.016

work page doi:10.1016/j.spa.2018.03.016 2016

[13] [13]

Huang, Y

J. Huang, Y . Jiaqiang, and Y . Shi (2020). Solvability of anticipated backward stochastic V olterra integral equations.Institute for Financial Studies and School of Mathematics, Shan- dong University

work page 2020

[14] [14]

Malliavin (1976)

P. Malliavin (1976). Stochastic calculus of variations and hypoelliptic operators. InProc. Internat. Symposium on Stochastic Differential Equations, Kyoto Univ., Kyoto, Wiley

work page 1976

[15] [15]

Agram, B

N. Agram, B. Øksendal, and S. Yakhlef (2018). New approach to optimal control of stochastic V olterra integral equations.Stochastics, DOI:10.1080/17442508.2018.1557186, pp. 1-25

work page doi:10.1080/17442508.2018.1557186 2018

[16] [16]

Nualart (2006).The Malliavin Calculus and Related Topics

D. Nualart (2006).The Malliavin Calculus and Related Topics. Springer

work page 2006

[17] [17]

Øksendal and A

B. Øksendal and A. Sulem (2001). A maximum principle for optimal control of stochastic systems with delay with applications to finance. InOptimal Control and PDEs, IOS Press, pp. 64-79. 23

work page 2001

[18] [18]

and Zabczyk, J

Peszat, S. and Zabczyk, J. (2008). Stochastic Partial Differential Equations with Lévy Noise (Encyclopedia Math. Appl. 113). Cambridge University Press

work page 2008

[19] [19]

Peng and Z

S. Peng and Z. Yang (2009). Anticipated backward stochastic differential equations.Annals of Probability, 37, 877-902

work page 2009

[20] [20]

Ren (2010)

Y . Ren (2010). On solutions of backward stochastic V olterra integral equations with jumps in Hilbert spaces.Journal of Optimization Theory and Applications, 144(2), 319-333

work page 2010

[21] [21]

Solvability of general backward stochastic V olterra integral equa- tions

Shi, Y ., Wang, T., 2012. Solvability of general backward stochastic V olterra integral equa- tions. J. Korean Math. Soc. 49, 1301-1321

work page 2012

[22] [22]

Sanz-Solé (2005).Malliavin Calculus with Applications to Stochastic Partial Differential Equations

M. Sanz-Solé (2005).Malliavin Calculus with Applications to Stochastic Partial Differential Equations. EPFL Press, Lausanne

work page 2005

[23] [23]

H. Wang, J. Sun, and J. Yong (2018). Quadratic backward stochastic V olterra integral equa- tions. arXiv:1810.10149

work page arXiv 2018

[24] [24]

Wen and Y

J. Wen and Y . Shi (2020). Solvability of anticipated backward stochastic V olterra integral equations.Statist. Probab. Lett., 122, 118-127

work page 2020

[25] [25]

Wen and Y

J. Wen and Y . Shi (2019). Symmetrical martingale solutions of backward doubly stochastic V olterra integral equations. arXiv:1909.04292

work page arXiv 2019

[26] [26]

H. Wu, W. Wang, and J. Ren (2012). Anticipated backward stochastic differential equations with non-Lipschitz coefficients.Statist. Probab. Lett., 82, 672-682

work page 2012

[27] [27]

Yong (2006)

J. Yong (2006). Backward stochastic V olterra integral equations and some related problems. Stochastic Processes and their Applications, 116(5), 779-795

work page 2006

[28] [28]

Yong (2008)

J. Yong (2008). Well-posedness and regularity of backward stochastic V olterra integral equa- tions.Probability Theory and Related Fields, 142, 21-77. 24

work page 2008