Anticipated backward stochastic Volterra integral equations and their applications to nonzero-sum stochastic differential games

Bixuan Yang; Tiexin Guo

arxiv: 2501.14263 · v7 · submitted 2025-01-24 · 🧮 math.PR

Anticipated backward stochastic Volterra integral equations and their applications to nonzero-sum stochastic differential games

Bixuan Yang , Tiexin Guo This is my paper

Pith reviewed 2026-05-23 04:54 UTC · model grok-4.3

classification 🧮 math.PR

keywords anticipated backward stochastic Volterra integral equationsstochastic differential gamesmaximum principleNash equilibriumMalliavin calculusstochastic delay Volterra integral equationslinear-quadratic games

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The pith

Linear anticipated BSVIEs serve as adjoint equations to derive the first maximum principle for nonzero-sum games of stochastic delay Volterra equations.

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

The paper extends the class of anticipated backward stochastic Volterra integral equations so that the generator incorporates both pointwise and average time-advanced functions. It proves well-posedness, comparison theorems, and regularity of adapted M-solutions via Malliavin calculus. These equations are then applied as adjoints in a nonzero-sum differential game whose state evolves according to stochastic delay Volterra integral equations, yielding a maximum principle and, in the linear-quadratic case, an explicit Nash equilibrium.

Core claim

By treating linear anticipated BSVIEs whose generators contain both pointwise and average time-advanced terms as the adjoint equations, the authors obtain the maximum principle for the nonzero-sum differential game problem driven by stochastic delay Volterra integral equations; the same principle produces a Nash equilibrium point for the corresponding linear-quadratic game.

What carries the argument

The linear anticipated BSVIE with mixed pointwise and average time-advanced generator, used as the adjoint process to characterize optimality in the SDVIE game.

If this is right

The maximum principle applies directly to nonzero-sum games whose dynamics are given by stochastic delay Volterra integral equations.
A Nash equilibrium point exists and can be characterized explicitly for the linear-quadratic version of the same game.
Well-posedness and comparison results hold for the enlarged class of anticipated BSVIEs that include both types of time-advanced terms.
Malliavin regularity results for adapted M-solutions extend previous work on standard BSVIEs.

Where Pith is reading between the lines

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

The same adjoint technique could be tested on games whose state equations contain nonlinear Volterra kernels.
Numerical schemes for solving the anticipated BSVIE might be used to compute approximate Nash controls in concrete delay games.
The comparison theorem for these BSVIEs may allow ordering of value functions across different game parameters.

Load-bearing premise

The generator of the anticipated BSVIE includes both pointwise and average time-advanced functions, and the required Lipschitz and integrability conditions hold so that Malliavin calculus applies to the solutions.

What would settle it

An explicit SDVIE game in which the candidate control obtained from the adjoint BSVIE fails to satisfy the Nash condition, or a generator with both pointwise and average terms for which no adapted M-solution exists.

read the original abstract

In [J. Wen, Y. Shi, Stat. Probab. Lett. 156 (2020) 108599] the authors first introduced a kind of anticipated backward stochastic Volterra integral equations (anticipated BSVIEs, for short). By virtue of the duality principle, it is found in this paper that the anticipated BSVIEs can be applied to the study of stochastic differential games. Naturally, in order to develop the related theories and applications of BSVIEs, in this paper we deeply investigate a more general class of anticipated BSVIEs whose generator includes both pointwise and average time-advanced functions. In theory, the well-posedness and the comparison theorem of anticipated BSVIEs are established, and some regularity results of adapted M-solutions are proved by applying Malliavin calculus, which cover the previous results for BSVIEs. Further, using linear anticipated BSVIEs as the adjoint equation, we present the maximum principle for the nonzero-sum differential game system of stochastic delay Volterra integral equations (SDVIEs, for short) for the first time. As one of the applications of the principle, a Nash equilibrium point of the linear-quadratic differential game problem of SDVIEs is obtained.

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

Extends anticipated BSVIEs to include averaged time-advanced terms and derives the first maximum principle for nonzero-sum SDVIE games using the linear case as adjoint.

read the letter

The main takeaway is that this paper adds an averaged time-advanced argument to the generator of anticipated BSVIEs, proves well-posedness and comparison under standard Lipschitz conditions, obtains Malliavin regularity, and then uses the linear version to state a maximum principle for nonzero-sum games whose dynamics are stochastic delay Volterra integral equations, with an explicit Nash point in the LQ case. It reduces to the 2020 Wen-Shi result when the average term is removed. The work carries the usual duality argument through to the new generator and the Volterra kernel without visible gaps once the variation process is adjusted for the delay structure. The LQ extraction follows the standard route and supplies a concrete example. The soft spot is that the abstract and stress-test give no explicit Lipschitz constants or measurability verification for the averaged term inside the Malliavin estimates; if those estimates hold without extra restrictions, the claims are fine, but a referee will need to check the details. The circularity burden looks low because nothing is fitted to data and the derivations rest on existing stochastic analysis tools. This is for readers already working on backward stochastic Volterra equations or delay games in control and finance. A specialist who needs the adjoint for nonzero-sum SDVIE problems will get direct value from the maximum principle and the LQ solution. The paper is coherent on its own terms and deserves a serious referee even though the advance is incremental rather than transformative.

Referee Report

2 major / 2 minor

Summary. The manuscript extends anticipated backward stochastic Volterra integral equations (anticipated BSVIEs) to the case where the generator incorporates both pointwise and averaged time-advanced arguments. It claims to prove well-posedness and a comparison theorem under Lipschitz-plus-integrability hypotheses, establish Malliavin regularity of the adapted solutions, and then use the linear anticipated BSVIE as adjoint to derive a maximum principle for nonzero-sum differential games driven by stochastic delay Volterra integral equations (SDVIEs), together with an explicit Nash equilibrium for the associated linear-quadratic problem.

Significance. If the derivations hold, the work supplies the first maximum principle for nonzero-sum SDVIE games and a concrete LQ Nash-equilibrium extraction, thereby extending the reach of anticipated BSVIE theory to control problems with both delay and Volterra memory. The Malliavin-regularity step is a standard but useful technical ingredient that covers earlier BSVIE results.

major comments (2)

[well-posedness and comparison theorem] The well-posedness and comparison results (abstract and the section establishing existence) are stated under 'Lipschitz and integrability conditions' without explicit constants or verification that the averaged time-advanced term preserves the measurability required for the subsequent Malliavin calculus and adjoint process; this is load-bearing for the maximum-principle claim.
[maximum principle for SDVIE games] The duality argument that identifies the linear anticipated BSVIE as the adjoint for the SDVIE game variation process is asserted but not sketched; the Volterra kernel and delay structure must be shown to produce no extra terms in the first-order variation (Section on the maximum principle).

minor comments (2)

Notation distinguishing the pointwise versus averaged advance operators should be introduced once and used consistently to improve readability.
[abstract] The abstract and introduction would benefit from a one-sentence statement of the precise Lipschitz constants employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments. We address the major points below and will revise the manuscript accordingly to enhance clarity and completeness.

read point-by-point responses

Referee: [well-posedness and comparison theorem] The well-posedness and comparison results (abstract and the section establishing existence) are stated under 'Lipschitz and integrability conditions' without explicit constants or verification that the averaged time-advanced term preserves the measurability required for the subsequent Malliavin calculus and adjoint process; this is load-bearing for the maximum-principle claim.

Authors: We agree that explicit constants improve readability. In the revision we will state the well-posedness and comparison theorems with concrete Lipschitz constants and integrability bounds. The averaged time-advanced term is defined via integration against Lebesgue measure on a deterministic interval; we will add a short remark confirming that this construction preserves adaptedness and the required measurability for Malliavin differentiability, thereby supporting the subsequent adjoint construction. revision: yes
Referee: [maximum principle for SDVIE games] The duality argument that identifies the linear anticipated BSVIE as the adjoint for the SDVIE game variation process is asserted but not sketched; the Volterra kernel and delay structure must be shown to produce no extra terms in the first-order variation (Section on the maximum principle).

Authors: We acknowledge the need for an explicit sketch. In the revised version we will expand the maximum-principle section with a step-by-step derivation of the first-order variation of the SDVIE state. Using the specific form of the Volterra kernel and the delay structure, together with integration-by-parts against the anticipated BSVIE adjoint, we will verify that no additional variation terms appear, thereby justifying the duality. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper extends the anticipated BSVIE framework introduced in Wen-Shi (2020) by considering generators with both pointwise and averaged time-advanced terms. Well-posedness, comparison, and Malliavin regularity are established under standard Lipschitz-plus-integrability hypotheses that do not presuppose the target maximum principle. The linear anticipated BSVIE is then invoked as adjoint via the duality principle, which is an external stochastic-analysis tool, to obtain the maximum principle for nonzero-sum SDVIE games and the LQ Nash equilibrium. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology; all load-bearing arguments rest on independently verifiable analytic estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper relies on standard assumptions of stochastic analysis (existence of Malliavin derivatives, adapted solutions in appropriate Banach spaces) without introducing new free parameters or invented entities.

axioms (2)

domain assumption Standard Lipschitz and linear growth conditions on the generator suffice for well-posedness of anticipated BSVIEs
Invoked to establish existence and uniqueness in the theory section of the abstract
domain assumption Malliavin calculus applies to adapted M-solutions of the generalized BSVIE
Used to obtain regularity results covering previous BSVIE cases

pith-pipeline@v0.9.0 · 5757 in / 1363 out tokens · 24148 ms · 2026-05-23T04:54:14.525910+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

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Abi Jaber, E

E. Abi Jaber, E. Miller, H. Pham, Markowitz portfolio selection for multivariate aﬃne and quadratic Volterra models, SIAM J. Financial Math. 12 (1) (202 1) 369–409

work page

[2] [2]

Agram, S

N. Agram, S. Haadem, B. Øksendal, F. Proske, A maximum principle for inﬁnite horizon delay equations, SIAM J. Math. Anal. 45 (2013) 2499–2522

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Agram, Y

N. Agram, Y. Hu, B. Øksendal, Mean-ﬁeld backward stochastic d iﬀerential equations and applications, Systems Control Lett. 162 (2022) 105196

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Buckdahn, J

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Douissi, J

S. Douissi, J. Wen, Y. Shi, Mean-ﬁeld anticipated BSDEs driven by f ractional Brownian motion and related stochastic control problem, Appl. Math. Compu t. 355 (2019) 282– 298

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S. Hamad` ene, R. Mu, Discontinuous Nash equilibrium points for no nzero-sum stochastic diﬀerential games, Stoch. Process. Appl. 130 (11) (2020) 6901– 6926

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Y. Hu, J. Huang, W. Li, Backward stochastic diﬀerential equat ions with conditional reﬂection and related recursive optimal control problems, SIAM J . Control Optim. 62 (5) (2024) 2557–2589

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Moon, A feedback Nash equilibrium for aﬃne-quadratic zero- sum stochastic diﬀer- ential games with random coeﬃcients, IEEE Control Syst

J. Moon, A feedback Nash equilibrium for aﬃne-quadratic zero- sum stochastic diﬀer- ential games with random coeﬃcients, IEEE Control Syst. Lett. 4 (4) (2020) 868–873

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Nualart, W

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Savku, G

E. Savku, G. Weber, Stochastic diﬀerential games for optimal investment problems in a Markov regime-switching jump-diﬀusion market, Ann. Oper. Res . 312 (2) (2022) 1171–1196

work page 2022

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T. Siu, Y. Shen, Risk-minimizing pricing and Esscher transform in a general non- Markovian regime-switching jump-diﬀusion model, Discrete Contin. D yn. Syst., Ser. B 22 (7) (2017) 2595–2626

work page 2017

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Y. Tian, J. Guo, Z. Sun, Optimal mean-variance reinsurance in a ﬁnancial market with stochastic rate of return, J. Ind. Manag. Optim. 17 (4) (2021) 1 887–1912

work page 2021

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G. Wang, H. Xiao, J. Xiong, A kind of LQ non-zero sum diﬀerential game of backward stochastic diﬀerential equation with asymmetric information, Auto matica 97 (2018) 346–352

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Wang, Extended backward stochastic Volterra integral eq uations, quasilinear parabolic equations, and Feynman-Kac formula, Stoch

H. Wang, Extended backward stochastic Volterra integral eq uations, quasilinear parabolic equations, and Feynman-Kac formula, Stoch. Dyn. 21 (1 ) (2021) 2150004

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H. Wang, J. Sun, J. Yong, Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations, App l. Math. Optim. 84 (2021) 145–190

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H. Wang, J. Yong, C. Zhou, Optimal controls for forward-bac kward stochastic diﬀeren- tial equations: Time-inconsistency and time-consistent solutions, J. Math. Pure. Appl. 190 (2024) 103603. 39

work page 2024

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Wang, Maximum principle for optimal control of mean-ﬁeld bac kward doubly SDEs with delay, J

M. Wang, Maximum principle for optimal control of mean-ﬁeld bac kward doubly SDEs with delay, J. Optim. Theory Appl. 205 (2025) 2

work page 2025

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Wang, Backward stochastic Volterra integro-diﬀerential e quations and applications in optimal control problems, SIAM J

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T. Wang, J. Yong, Comparison theorems for some backward st ochastic Volterra integral equations, Stoch. Process. Appl. 125 (5) (2015) 1756–1798

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T. Wang, J. Yong, Backward stochastic Volterra integral equ ations–Representation of adapted solutions, Stoch. Process. Appl. 129 (12) (2019) 4926– 4964

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T. Wang, J. Yong, Spike variations for stochastic Volterra inte gral equations, SIAM J. Control Optim. 61 (6) (2023) 3608–3634

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T. Wang, M. Zheng, Singular backward stochastic Volterra inte gral equations in inﬁnite dimensional spaces, J. Diﬀerential Equations 407 (2024) 1–56

work page 2024

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J. Wen, Y. Shi, Anticipative backward stochastic diﬀerential eq uations driven by frac- tional Brownian motion, Stat. Probab. Lett. 122 (2017) 118–127

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[35] [35]

J. Wen, Y. Shi, Solvability of anticipated backward stochastic Vo lterra integral equa- tions, Stat. Probab. Lett. 156 (2020) 108599

work page 2020

[36] [36]

F. Wu, J. Xiong, X. Zhang, Zero-sum stochastic linear-quadra tic Stackelberg diﬀerential games with jumps, Appl. Math. Optim. 89 (2024) 29

work page 2024

[37] [37]

F. Wu, G. Yin, Two-time-scale stochastic functional diﬀerentia l equations: Inclusion of inﬁnite delay and coupled segment processes, J. Diﬀerential Equ ations 435 (2025) 113238

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