Backward doubly stochastic Volterra integral equations and applications to optimal control problems

Jiaqiang Wen; Jie Xiong; Yufeng Shi

arxiv: 1906.10582 · v1 · pith:4UNWIVJMnew · submitted 2019-06-25 · 🧮 math.PR · math.OC

Backward doubly stochastic Volterra integral equations and applications to optimal control problems

Yufeng Shi , Jiaqiang Wen , Jie Xiong This is my paper

Pith reviewed 2026-05-25 16:04 UTC · model grok-4.3

classification 🧮 math.PR math.OC

keywords backward doubly stochastic Volterra integral equationsM-solutionscomparison theoremduality principlePontryagin maximum principleoptimal controlforward doubly stochastic Volterra integral equationsstochastic Volterra equations

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The pith

Backward doubly stochastic Volterra integral equations admit M-solutions, with a comparison theorem establishing existence for continuous coefficients and yielding a Pontryagin maximum principle for related optimal control problems.

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

The paper introduces backward doubly stochastic Volterra integral equations and defines M-solutions to study their well-posedness. It proves a comparison theorem that supports existence of solutions when the coefficients are continuous rather than Lipschitz. The same theorem underpins a duality principle linking linear forward and backward versions of these equations. This duality in turn produces a maximum principle for an optimal control problem driven by the forward equations.

Core claim

BDSVIEs are well-posed in the M-solution sense; a comparison theorem for these equations is established and then used to obtain existence when coefficients are continuous; a duality principle holds between linear FDSVIEs and BDSVIEs; and a Pontryagin-type maximum principle follows for the optimal control of FDSVIEs.

What carries the argument

The comparison theorem for BDSVIEs, which transfers existence from Lipschitz to continuous coefficients and supports the duality and maximum principle results.

If this is right

M-solutions exist and are unique for BDSVIEs under the stated conditions.
Existence holds for BDSVIEs whose coefficients are continuous.
A duality principle connects linear FDSVIEs to BDSVIEs.
A Pontryagin-type maximum principle governs optimal control problems for FDSVIEs.

Where Pith is reading between the lines

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

The comparison theorem may simplify verification of solution existence in applied models that naturally produce continuous but non-Lipschitz drivers.
The duality could be used to convert certain backward control problems into equivalent forward ones that are easier to simulate.
Extensions to equations with jumps or regime switches would require checking whether the comparison argument survives the added randomness.

Load-bearing premise

The coefficients must satisfy the integrability and measurability conditions required for the comparison theorem to apply.

What would settle it

An explicit BDSVIE with continuous coefficients that possesses no M-solution would falsify the existence claim derived from the comparison theorem.

read the original abstract

Backward doubly stochastic Volterra integral equations (BDSVIEs, for short) are introduced and studied systematically. Well-posedness of BDSVIEs in the sense of introduced M-solutions is established. A comparison theorem for BDSVIEs is proved. By virtue of the comparison theorem, we derive the existence of solutions for BDSVIEs with continuous coefficients. Furthermore, a duality principle between linear (forward) doubly stochastic Volterra integral equation (FDSVIE, for short) and BDSVIE is obtained. A Pontryagin type maximum principle is also established for an optimal control problem of FDSVIEs.

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 introduces BDSVIEs and M-solutions, proves well-posedness plus comparison, and derives a max principle, but the existence step for continuous coefficients rests on whether the comparison theorem's hypotheses are met by continuity alone.

read the letter

The paper defines backward doubly stochastic Volterra integral equations and introduces M-solutions to make sense of them. It shows well-posedness in that sense, proves a comparison theorem, and then claims that the comparison yields existence when coefficients are merely continuous. It also gets a duality between the linear forward and backward versions and a Pontryagin maximum principle for controlling the forward equations. These objects and the comparison theorem look new relative to the BSDE and Volterra results referenced in the abstract, and the control application follows naturally from the setup. The systematic treatment and the link to optimal control are the parts that hold up cleanly on the abstract. The soft spot is the existence claim for continuous coefficients. Comparison theorems in this area usually need Lipschitz conditions, monotonicity, or kernel integrability beyond plain continuity and measurability. The abstract presents the existence result as following directly from the comparison, but does not indicate whether the paper verifies that arbitrary continuous data satisfy those extra requirements or whether the M-solution definition supplies hidden regularity. Without the proofs it is impossible to tell how tight the argument is. This work is for people already working on stochastic Volterra equations or memory-dependent control problems. A reader who needs the comparison or the maximum principle would get concrete value from the theorems. It deserves a serious referee to check the comparison hypotheses and the existence derivation in detail.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces backward doubly stochastic Volterra integral equations (BDSVIEs) and establishes their well-posedness in the sense of M-solutions. It proves a comparison theorem for BDSVIEs, uses this to derive existence of solutions when coefficients are continuous, obtains a duality principle between linear forward doubly stochastic Volterra integral equations (FDSVIEs) and BDSVIEs, and establishes a Pontryagin-type maximum principle for an optimal control problem governed by FDSVIEs.

Significance. If the proofs are complete and the comparison theorem applies as claimed, the work introduces a new class of doubly stochastic Volterra equations with associated well-posedness theory, comparison results, duality, and control applications. The definition of M-solutions is a novel technical contribution that could support further developments in stochastic analysis and control.

major comments (1)

[Abstract] Abstract (and the existence claim derived from the comparison theorem): the derivation of existence for BDSVIEs with continuous coefficients via the comparison theorem is load-bearing for the central well-posedness result. Standard comparison theorems for backward stochastic Volterra equations require Lipschitz continuity or monotonicity in the state variable plus specific integrability on kernels and drivers; it is not shown that mere continuity and measurability suffice to meet these hypotheses, so the step from comparison to existence for arbitrary continuous data needs explicit verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (and the existence claim derived from the comparison theorem): the derivation of existence for BDSVIEs with continuous coefficients via the comparison theorem is load-bearing for the central well-posedness result. Standard comparison theorems for backward stochastic Volterra equations require Lipschitz continuity or monotonicity in the state variable plus specific integrability on kernels and drivers; it is not shown that mere continuity and measurability suffice to meet these hypotheses, so the step from comparison to existence for arbitrary continuous data needs explicit verification.

Authors: We agree that an explicit verification of the applicability of the comparison theorem to continuous coefficients would improve the clarity of the argument. The comparison theorem (proved in Section 3) is established within the M-solution framework, which is designed to accommodate weaker regularity than the classical Lipschitz setting. Nevertheless, to make the passage from the comparison result to existence fully transparent, we will add a short remark or paragraph immediately after the comparison theorem that directly checks the integrability and measurability conditions for continuous coefficients. This will be a minor but targeted addition. revision: yes

Circularity Check

0 steps flagged

No circularity: definitions precede independent analysis

full rationale

The paper first introduces BDSVIEs and M-solutions as new objects, then establishes well-posedness, proves a comparison theorem from those definitions, and applies the theorem to obtain existence for continuous coefficients. No step reduces by the paper's own equations to a fitted parameter, self-citation loop, or renamed input; the derivation chain is self-contained standard analysis without invoking the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on standard stochastic-analysis background and introduces one new solution concept; no free parameters or invented physical entities appear.

axioms (1)

standard math Standard filtered probability space supporting the requisite Brownian motions and Poisson measures for doubly stochastic integrals
Invoked implicitly when defining the integrals and M-solutions.

invented entities (1)

M-solutions no independent evidence
purpose: To serve as the notion of solution for which well-posedness of BDSVIEs is proved
New concept introduced in the paper; no independent evidence supplied beyond the definition itself.

pith-pipeline@v0.9.0 · 5634 in / 1380 out tokens · 53573 ms · 2026-05-25T16:04:39.523451+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 2 internal anchors

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Wen and Y

J. Wen and Y. Shi, Backward doubly stochastic diﬀerential equations with ran dom coeﬃcients and quasilinear stochastic PDEs, J. Math. Anal. Appl., 476 (2019), 86–100

work page 2019
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Wang and J

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

work page 2015
[34]

Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

T. Wang and J. Yong, Backward stochastic Volterra integral equations-represe ntation of adapted solu- tions, Arxiv:1802.03659

work page internal anchor Pith review Pith/arXiv arXiv
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Wang and H

T. Wang and H. Zhang. Optimal Control Problems of Forward-Backward Stochastic V olterra Integral Equations with Closed Control Regions, SIAM J. Control Optim., 55 (2017), 2574–2602

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Xiong, Super-Brownian motion as the unique strong solution to an SP DE, Ann

J. Xiong, Super-Brownian motion as the unique strong solution to an SP DE, Ann. Probab., 41 (2013), 1030–1054

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J. Yong, Backward stochastic Volterra integral equations and some r elated problems, Stoch. Proc. Appl., 116 (2006) 779–795

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[1] [1]

Optimal control of forward-backward stochastic Volterra equations

N. Agram, B. Oksendal and S. Yakhlef, Optimal control of forward-backward stochastic Volterra e qua- tions, Arxiv.org/abs/1606.03280

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Agram, B

N. Agram, B. Oksendal and S Yakhlef, New approach to optimal control of stochastic Volterra inte gral equations, Stochastics, (2018), 1–22

work page 2018

[3] [3]

V. Anh, W. Grecksch and J. Yong, Regularity of backward stochastic Volterra integral equat ions in Hilbert spaces, Stoch. Anal. Appl., 29 (2010), 146–168

work page 2010

[4] [4]

Artzner, F

P. Artzner, F. Delbaen, J.M. Eber and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), 203–228

work page 1999

[5] [5]

Boufoussi, J

B. Boufoussi, J. Casteren and N. Mrhardy, Generalized backward doubly stochastic diﬀerential equat ions and SPDEs with nonlinear Neumann boundary conditions, Bernoulli, 13 (2007), 423–446

work page 2007

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Bender and S

C. Bender and S. Pokalyuk, Discretization of backward stochastic Volterra integral e quations, Recent Developments in Computational Finance, Interdiscip. Math. Sci., 14 , World Sci. Publ., Hackensack, NJ, 2013, 245–278. 21

work page 2013

[7] [7]

Diehl and P

J. Diehl and P. Friz, Backward stochastic diﬀerential equations with rough driv ers, Ann. Probab., 40 (2012), 1715–1758

work page 2012

[8] [8]

Djordjevic and S

J. Djordjevic and S. Jankovic, On a class of backward stochastic Volterra integral equatio ns, Appl. Math. Lett., 26 (2013), 1192–1197

work page 2013

[9] [9]

Djordjevic and S

J. Djordjevic and S. Jankovic, Backward stochastic Volterra integral equations with addi tive perturba- tions, Appl. Math. Comput., 265 (2015), 903–910

work page 2015

[10] [10]

El Karoui and S

N. El Karoui and S. Huang, A general result of existence and uniqueness of backward sto chastic diﬀer- ential equations, Pitman Res. Notes Math. Ser., (1997), 27–38

work page 1997

[11] [11]

El Karoui, S

N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic diﬀerential equations in ﬁnance, Math. Finance, 7 (1997), 1–71

work page 1997

[12] [12]

Gianin, Risk measures via g-expectations, Insurance: Mathematics and Economics, 39 (2006), 19–34

E.R. Gianin, Risk measures via g-expectations, Insurance: Mathematics and Economics, 39 (2006), 19–34

work page 2006

[13] [13]

Hu and B

Y. Hu and B. Oksendal, Linear Volterra backward stochastic integral equations, Stoch. Proc. Appl., 129 (2019), 626–633

work page 2019

[14] [14]

Y. Han, S. Peng and Z. Wu, Maximum Principle for Backward Doubly Stochastic Control S ystems with Applications, SIAM J. Control Optim., 48 (2010), 4224-4241

work page 2010

[15] [15]

Kromer and L

E. Kromer and L. Overbeck, Classical diﬀerentiability of BSVIEs and dynamic capital a llocations, International Journal of Theoretical and Applied Finance, 20 (2017), 1–26

work page 2017

[16] [16]

Lin, Adapted solution of a backward stochastic nonlinear Volter ra integral equations, Stoch

J. Lin, Adapted solution of a backward stochastic nonlinear Volter ra integral equations, Stoch. Anal. Appl., 20 (2002), 165–183

work page 2002

[17] [17]

Lepeltier and J

J. Lepeltier and J. Martin, Backward stochastic diﬀerential equations with continuou s coeﬃcient, Stat. Probab. Lett., 32 (1997), 425–430

work page 1997

[18] [18]

Ma and J

J. Ma and J. Yong, Forward-Backward Stochastic Diﬀerential Equations and Their Ap plications, Springer-Verlag, Berlin, 1999

work page 1999

[19] [19]

Overbeck and J.A.L

L. Overbeck and J.A.L. Roder, Path-dependent backward stochastic Volterra integral equ ations with jumps, diﬀerentiability and duality principle, Probability, Uncertainty and Quantitative Risk, 3 (2018), 4

work page 2018

[20] [20]

Pardoux and S

E. Pardoux and S. Peng, Adapted solution of a backward stochastic diﬀerential equa tion, Syst. Control. Lett., 14 (1990), 55–61

work page 1990

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Pardoux and S

E. Pardoux and S. Peng, Backward stochastic diﬀerential equations and quasilinea r parabolic partial diﬀerential equations, In: B. L. Rozuvskii and R. B. Sowers (eds.) Stochastic partial diﬀerential equations and their applications, Lect. Notes Control Inf. Sci., vol. 176, 200–217, Berlin Heidelberg New York: Springer. 1992

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Pardoux and S

E. Pardoux and S. Peng, Backward doubly stochastic diﬀerential equations and syst ems of quasilinear SPDEs, Probab. Theory Relat. Fields, 98 (1994), 209–227

work page 1994

[23] [23]

Peng, Probabilistic Interpretations for Systems of Quasilinear Parabolic Partial Diﬀerential Equation, Stochastic, 37 (1991), 61–74

S. Peng, Probabilistic Interpretations for Systems of Quasilinear Parabolic Partial Diﬀerential Equation, Stochastic, 37 (1991), 61–74. 22

work page 1991

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Peng, Nonlinear expectations, nonlinear evaluations and risk me asures, Stochastic methods in ﬁnance

S. Peng, Nonlinear expectations, nonlinear evaluations and risk me asures, Stochastic methods in ﬁnance. Springer Berlin Heidelberg, 1856 (2004), 165–253

work page 2004

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Y. Shi, Y. Gu and K. Liu, Comparison Theorems of Backward Doubly Stochastic Diﬀeren tial Equations and Applications, Anal. Appl., 23 (2005), 97-110

work page 2005

[26] [26]

Y. Shi, J. Wen and J. Xiong, Forward-Backward Doubly Stochastic Volterra Integral Equ ations and Some Related Problems, working paper

work page

[27] [27]

Y. Shi, T. Wang, Solvability of general backward stochastic Volterra integ ral equations, J. Korean Math. Soc., 49 (2012) 1301–1321

work page 2012

[28] [28]

Y. Shi, T. Wang and J. Yong, Mean-ﬁeld backward stochastic Volterra integral equation s, Dis. Cont. Dyn. System Ser. B, 18 (2013), 1929–1967

work page 2013

[29] [29]

Y. Shi, T. Wang and J. Yong, Optimal Control Problems of Forward-Backward Stochastic V olterra Integral Equations, Math. Control Related Fields, 5 (2015), 613–649

work page 2015

[30] [30]

H. Wang, J. Sun and J. Yong, Quadratic Backward Stochastic Volterra Integral Equation s, Arxiv.org/abs/1810.10149

work page arXiv

[31] [31]

Wang, Linear quadratic control problems of stochastic Volterra i ntegral equations, ESAIM: COCV, to appear

T. Wang, Linear quadratic control problems of stochastic Volterra i ntegral equations, ESAIM: COCV, to appear

work page

[32] [32]

Wen and Y

J. Wen and Y. Shi, Backward doubly stochastic diﬀerential equations with ran dom coeﬃcients and quasilinear stochastic PDEs, J. Math. Anal. Appl., 476 (2019), 86–100

work page 2019

[33] [33]

Wang and J

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

work page 2015

[34] [34]

Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

T. Wang and J. Yong, Backward stochastic Volterra integral equations-represe ntation of adapted solu- tions, Arxiv:1802.03659

work page internal anchor Pith review Pith/arXiv arXiv

[35] [35]

Wang and H

T. Wang and H. Zhang. Optimal Control Problems of Forward-Backward Stochastic V olterra Integral Equations with Closed Control Regions, SIAM J. Control Optim., 55 (2017), 2574–2602

work page 2017

[36] [36]

Xiong, Super-Brownian motion as the unique strong solution to an SP DE, Ann

J. Xiong, Super-Brownian motion as the unique strong solution to an SP DE, Ann. Probab., 41 (2013), 1030–1054

work page 2013

[37] [37]

Yong, Backward stochastic Volterra integral equations and some r elated problems, Stoch

J. Yong, Backward stochastic Volterra integral equations and some r elated problems, Stoch. Proc. Appl., 116 (2006) 779–795

work page 2006

[38] [38]

Yong, Continuous-time dynamic risk measures by backward stochas tic Volterra integral equations, Appl

J. Yong, Continuous-time dynamic risk measures by backward stochas tic Volterra integral equations, Appl. Anal., 86 (2007), 1429–1442

work page 2007

[39] [39]

Yong, Well-posedness and regularity of backward stochastic Volt erra integral equations, Probab

J. Yong, Well-posedness and regularity of backward stochastic Volt erra integral equations, Probab. The- ory Relat. Fields, 142 (2008), 21–77

work page 2008

[40] [40]

Yong and X

J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999. 23

work page 1999