Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Optimal Control of Heterogeneous Mean-Field Stochastic Differential Equations with Common Noise and Applications

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper derives a closed-form optimal control for heterogeneous mean-field systems with common noise, via a new infinite-dimensional Riccati BSDE system.

desk verdict Good problem, plausible program, but the key square-completion identity in Prop 4.11 is algebraically wrong as printed, so the verification theorem and the main existence results are not established. read the letter →

arxiv 2511.18636 v2 pith:4OKD77GQ submitted 2025-11-23 math.OC math.PR

Filippo de Feo , Samy Mekkaoui This is my paper
classification math.OCmath.PR MSC 49N1093E2060K35
keywords mean-fieldSDEcommonnoiseheterogeneousagentslinear-quadraticcontrolbackwardstochasticRiccatiequationsinfinite-dimensionalHilbertspacesoptimaltradingsystemicrisk
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a linear-quadratic control problem for a continuum of heterogeneous agents coupled through their conditional means and sharing a common Brownian noise can be solved exactly. The solution is a triangular system of four backward stochastic Riccati equations on Hilbert spaces of square-integrable kernels: a label-wise Riccati BSDE, a new infinite-dimensional Riccati BSDE for the interaction kernel, and two linear BSDEs. The paper proves existence and uniqueness for this system and derives the optimal control in explicit feedback form, with the value function given by a quadratic-plus-linear-plus-constant expression. A sympathetic reader would care because this is the first treatment of heterogeneous mean-field control with common noise, and the explicit formulas are demonstrated on optimal trading and systemic risk models.

What carries the argument

The central machinery is the triangular system of backward stochastic Riccati equations (BSREs) on the Hilbert spaces L2(I;R^d) and L2(I×I;R^(d×d)), together with the fundamental relation obtained by completing the square. The first equation solves a standard per-agent Riccati BSDE for each label; the second is a new infinite-dimensional Riccati BSDE for the interaction kernel; the last two are linear BSDEs that absorb affine terms. The work they do is to make the cost functional a sum of a control-free term and a manifestly nonnegative quadratic term, so the zero of that term is the optimal feedback.

What would settle it

Solve the scalar per-agent Riccati BSDE explicitly for a family of coefficients satisfying the paper's assumptions and check whether the supremum over labels of the solution is finite. If a family with uniformly bounded coefficients can be found for which this supremum is infinite, the uniform bound fails and the proof of the verification theorem cannot be carried out; alternatively, an independent numerical solution of the full LQ problem for the two applications can be compared with the claimed feedback to confirm the formula.

Watch

Extended reading notes

Core claim

The paper's central claim is that the conditional cost functional admits a fundamental relation: it equals a function of the initial state alone plus an integral of a nonnegative quadratic form in the control, shifted by a linear function of the state and the conditional mean. This identity is proved by completing the square, using the four Riccati equations. Because the quadratic term vanishes exactly at the stated feedback control, the minimizer is unique; the corresponding value function is the explicit quadratic-linear-constant expression given in the paper. The existence and uniqueness of the infinite-dimensional backward Riccati equations is the technical heart, established by a per-la

Load-bearing premise

The argument that the per-agent Riccati BSDE solutions are uniformly bounded in the label is assumed rather than proved; if the essential supremum over labels can be infinite while all other assumptions hold, the infinite-dimensional system and the verification theorem collapse.

Editorial extensions

If this is right

  • If the central theorem is correct, the optimal control is given in closed loop by the feedback formula, so no numerical solution of the full control problem is required once the four Riccati equations are solved.
  • The value function separates into per-agent quadratic, kernel quadratic, linear, and constant terms, making the optimal cost computable from the four BSDE solutions.
  • The existence and uniqueness result covers a Riccati BSDE on an infinite-dimensional Hilbert space that standard theory does not handle, so the same triangular method applies to other LQ mean-field problems with common noise.
  • In the two financial applications, the theory yields explicit trading and lending/borrowing policies for heterogeneous agents facing common shocks.
  • The homogeneous and deterministic limits recover previously known Riccati ODE and mean-field results, providing consistency checks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the triangular structure suggests a practical numerical route—first solve the decoupled per-label Riccati BSDEs, then the kernel BSDE, then the two linear BSDEs—so the main computational obstacle is the infinite-dimensional kernel equation.
  • Editorial inference: a natural testable extension is a finite-N particle approximation of the continuum; the feedback control should be approximately optimal with an error governed by the regularity of the graphon kernels, though the paper does not compute such rates.
  • Editorial inference: the same framework should extend to vector-valued common noise and to controls entering both drift and diffusion, at the price of more complex Riccati coupling, since the argument only uses operator-norm and L2 structure.
  • Editorial inference: the load-bearing uniform bound on the per-agent Riccati solutions could be replaced by a label-wise bound plus a growth condition on the label, which would widen applicability to unbounded coefficient families.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops an LQ optimal control framework for heterogeneous mean-field SDEs with common noise, Eq. (3.1)-(3.6). The authors introduce a triangular system of backward Riccati equations: per-agent standard BSREs (4.1), an infinite-dimensional backward stochastic Riccati equation (4.4) on L2(I x I), a linear BSDE (4.8), and scalar linear BSDEs (4.11). The main theoretical results are a 'fundamental relation' (Prop. 4.11) decomposing the cost into a value function plus a nonnegative quadratic term, solvability theorems for the Riccati system, and a verification theorem (Thm. 4.15) identifying the optimal feedback control (4.23). Two financial applications are then presented: optimal trading with heterogeneous market participants and a systemic risk model with heterogeneous banks.

Significance. If correct, this would be a valuable first extension of heterogeneous mean-field control to settings with common noise, with a novel infinite-dimensional Riccati BSDE. The paper is clearly structured, transparent about the triangular nature of the Riccati system, and contains a careful treatment of label-measurability of conditional laws. However, the central proof of the fundamental relation contains an algebraic error, and the existence theory relies on an extra uniform-in-label bound that is assumed rather than proved. As written, the main theorem and the optimal feedback characterization are unsupported.

major comments (3)
  1. [Appendix B, Eq. (B.4)-(B.5)] The square-completion identity for chi_t is algebraically wrong. For chi(alpha)=2<zeta,alpha>+<alpha,O alpha> with zeta=UX+int V \bar X dv+Gamma, the correct subtraction is <zeta,O^{-1}zeta> = <UX,O^{-1}UX> + 2<UX,O^{-1}V\bar X> + 2<UX,O^{-1}Gamma> + <V\bar X,O^{-1}V\bar X> + 2<V\bar X,O^{-1}Gamma> + <Gamma,O^{-1}Gamma>. Display (B.4) instead writes -int <UX,O^{-1}V\bar X> - int <V\bar X,O^{-1}U\bar X> (with \bar X in place of X in the second term) and -int <\bar X,V(v,u)^T O_v^{-1}Gamma_v> - <int V(v,u)^T O_v^{-1}Gamma_v, X> (X and \bar X swapped). Hence the X-\bar X cross term is incorrect and an extraneous \bar X-\bar X term is created; in (B.5) the required -(V^*O^{-1}V) quadratic term is missing from the \bar K bracket. The cancellation with F in (4.4) therefore fails, Proposition 4.11 is not proved, and Theorem 4.15 together with the optimal feedback (4.23), both resting on (4.14),
  2. [Section 4.2.1, Eq. (4.16)] The bound sup_t |K^u_t| <= C, du x dP-a.e., is not a consequence of Assumption 3.1. The standard per-label solvability invoked above (4.16) yields constants depending on the u-labeled coefficients A^u, B^u, C^u, D^u, Q^u, R^u, H^u; since the label set I is uncountable, these constants need not be uniformly bounded. The manuscript simply assumes 'that the essential supremum with respect to the label is also finite.' This uniform bound is load-bearing: it is used to place (K,Z^K) in S^2_{F0}(I;R^{d x d}) x H^2_{F0}(I;R^{d x d}) and to justify the Borel representation in Step 2. The whole triangular system (4.4)-(4.11) and Theorem 4.15 depend on it. A proof, or an explicit uniform-in-label controllability-type assumption, must be supplied.
  3. [Section 4.2.2, Props. 4.12-4.13] The a priori estimate (4.19) and the contraction argument in Proposition 4.13 use the fundamental relation (4.14) as the mechanism for controlling ||T_{\bar K_t}||. Since (4.14) rests on the invalid square completion identified above, the existence and uniqueness proof for (4.4) does not follow as written. In particular, the lower bound in (4.21) is obtained from (4.14) by 'taking the optimal control setting the quadratic term to be null', and the exact form of that relation is essential. The two defects together leave the solvability of the abstract Riccati equation open.
minor comments (4)
  1. [Section 2] The parenthetical remark 'Not sure if I used them after' should be removed or resolved: either the spaces S^2_{\tilde F0}(I x I;E), H^2_{\tilde F0}(I x I;E) are used and should be formally introduced, or they are not and the sentence should be deleted.
  2. [Sections 5.1-5.2] The coefficient lists for the applications are difficult to verify: in Section 5.1 the expression for M_t(u) runs into the definition of Gamma_t^u without a separator, and the integral in Lambda_t^u is written as \int_U although U is not defined there; Section 5.2 has similar formatting problems. These displays should be rewritten with clear punctuation.
  3. [Remark 3.8] The displayed formula for G^H(u,v) is written with unbalanced parentheses and the claimed equivalence with the centered formulation (3.10) is not immediately verifiable. Please correct the display and check the expansion, especially the cross terms.
  4. [Throughout] There are several typos: the abstract has 'in term of'; Remark 4.10(iv) says 'it will clear from the proof'; Appendix A contains mismatched subscripts in sup expressions (e.g., \bar X^{Y,u}_q versus \bar X^{Y,u}_s). These are presentation issues only.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Riccati system and verification theorem are genuine derivations; self-citations are methodological rather than load-bearing.

full rationale

The derivation chain is (3.1)-(3.6) → the Riccati system (4.1)-(4.11) → the fundamental relation (4.14) → the feedback control (4.23) and verification Theorem 4.15. The Riccati equations are introduced as objects to be solved, not fitted to the cost functional, and the control is expressed through their solutions. The fundamental relation is obtained by Itô's formula and a square-completion argument; this is the standard non-circular way in which Riccati equations are constructed, since the driver F in (4.4)-(4.5) is chosen so that the drift terms cancel. Existence of \bar K is proved by contraction on a terminal interval followed by continuation using the a priori estimate (4.19), which is stated for any solution on a terminal interval; this is a standard continuation argument, not a circular one. The uniform bound (4.16) for K is justified by per-agent standard BSRE theory together with the explicitly stated additional assumption that the label-essential supremum is finite; this is a stated hypothesis, not a disguised reuse of the conclusion. The main self-citations ([24], [45]) supply techniques such as Picard schemes and Borel-measurable representations; they are not cited as an external uniqueness theorem that forces the paper's choice, so they do not constitute load-bearing circular support. A separate concern — the square-completion identity in (B.4) appears algebraically incorrect, since the X–\bar X cross term is printed with \bar X in both factors and the \bar K bracket in (B.5) then fails to cancel — is a correctness/missing-proof issue rather than circularity: if true, the fundamental relation would fail, not be equivalent to its inputs. Accordingly, the paper shows no significant circularity; the score of 1 reflects only the methodological reliance on the authors' earlier work.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The main imported ingredients are standard BSDE theory and a uniform-in-label bound that is assumed rather than derived. The ad hoc assumption (4.16) is the most fragile item in the ledger.

assumptions (5)
  • domain assumption Assumption 3.1: L∞/S² regularity of model coefficients (β,γ,θ,A,C,GA,GC,B,D).
    Needed for well-posedness of the state equation (3.1) and for the BSDE system.
  • domain assumption Assumption 3.6: Nonnegativity (3.8) and uniform lower bound R >= c I.
    Ensures the cost functional is well defined and the Riccati feedback is coercive.
  • ad hoc to paper Uniform-in-label bound for the standard Riccati solution K (Eq. (4.16)).
    Assumed without proof in Section 4.2.1; not a consequence of Assumption 3.1. Needed for measurability and integrability of (4.1).
  • standard math Standard BSDE/BSRE existence and uniqueness results ([52], [4], [33]).
    Used to solve the per-u Riccati equation (4.1) and linear BSDEs (4.8), (4.11).
  • domain assumption Measurability of conditional laws w.r.t. the common noise (Definition 3.2/Remark 3.3).
    The well-posedness arguments rely on joint measurability of u ↦ E[X^u | F0]; this is imposed via representation assumptions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal Control of Heterogeneous Mean-Field Stochastic Differential Equations with Common Noise and Applications." pith.science (2026). https://pith.science/paper/4OKD77GQ

@misc{pith2026251118636,
  author       = {Pith},
  title        = {Pith review of: Optimal Control of Heterogeneous Mean-Field Stochastic Differential Equations with Common Noise and Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OKD77GQ}},
  note         = {Machine review of arXiv:2511.18636}
}
read the original abstract

We initiate the study of optimal control problems of heterogeneous mean-field stochastic differential equations with common noise. We formulate the problem within a linear-quadratic framework, a particularly important class in control theory, typically renowned for its analytical tractability and broad range of applications. We derive a novel system of backward stochastic Riccati equations on infinite-dimensional Hilbert spaces. As this system is not covered by standard theory, we establish existence and uniqueness of solutions. We explicitly characterize the optimal control in terms of the solution of this system. We apply these results to solve two problems arising in mathematical finance: optimal trading with heterogeneous market participants and systemic risk in networks of heterogeneous banks.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Infinite-Dimensional LQ Mean Field Games with Common Noise: Small and Arbitrary Finite Time Horizons

    math.OC 2026-01 conditional novelty 6.0 of 10

    Infinite-dimensional linear-quadratic mean field games with common noise have unique equilibria for small time horizons and, under deterministic common-noise diffusion, for arbitrary finite time horizons.

Reference graph

Works this paper leans on

66 extracted references · 8 linked inside Pith · cited by 1 Pith paper

  1. [24]

    Linear-quadratic optimal control for non-exchangeable mean-field sdes and applications to systemic risk.arXiv preprint arXiv:2503.03318, 2025

    Anna De Crescenzo, Filippo de Feo, and Huyên Pham. Linear-quadratic optimal control for non-exchangeable mean-field sdes and applications to systemic risk.arXiv preprint arXiv:2503.03318, 2025

    arXiv 2025

  2. [45]

    Stochastic maximum principle for optimal control problem of non- exchangeable mean field systems.arXiv preprint arXiv:2506.05595, 2025

    Idris Kharroubi, Samy Mekkaoui, and Huyên Pham. Stochastic maximum principle for optimal control problem of non- exchangeable mean field systems.arXiv preprint arXiv:2506.05595, 2025

    arXiv 2025

  3. [1]

    Optimal execution of portfolio transactions.Journal of Risk, 3:5–40, 2001

    Robert Almgren and Neil Chriss. Optimal execution of portfolio transactions.Journal of Risk, 3:5–40, 2001

    2001

  4. [2]

    Contracting a crowd of heterogeneous agents.arXiv preprint arXiv:2507.09415, 2025

    Guillermo Alonso Alvarez, Erhan Bayraktar, and Ibrahim Ekren. Contracting a crowd of heterogeneous agents.arXiv preprint arXiv:2507.09415, 2025

    arXiv 2025

  5. [3]

    Stochastic graphon games: Ii

    Alexander Aurell, René Carmona, and Mathieu Laurière. Stochastic graphon games: Ii. the linear-quadratic case.Applied Mathematics & Optimization, 85(3):39, 2022

    2022

  6. [4]

    A weak martingale approach to linear-quadratic mckean–vlasov stochastic control problems

    Matteo Basei and Huyên Pham. A weak martingale approach to linear-quadratic mckean–vlasov stochastic control problems. Journal of Optimization Theory and Applications, 181(2):347–382, 2019

    2019

  7. [5]

    Graphon mean field systems.The Annals of Applied Probability, 33(5):3587–3619, 2023

    Erhan Bayraktar, Suman Chakraborty, and Ruoyu Wu. Graphon mean field systems.The Annals of Applied Probability, 33(5):3587–3619, 2023

    2023

  8. [6]

    Graphon particle system with common noise.arXiv preprint arXiv:2507.03265, 2025

    Erhan Bayraktar, Xihao He, and Donghan Kim. Graphon particle system with common noise.arXiv preprint arXiv:2507.03265, 2025

    arXiv 2025

Show all 66 references
  1. [7]

    Concentration of measure for graphon particle system.Advances in Applied Probability, 56(4):1279–1306, 2024

    Erhan Bayraktar and Donghan Kim. Concentration of measure for graphon particle system.Advances in Applied Probability, 56(4):1279–1306, 2024

    2024

  2. [8]

    Stationarity and uniform in time convergence for the graphon particle system.Stochastic Processes and their Applications, 150:532–568, 2022

    Erhan Bayraktar and Ruoyu Wu. Stationarity and uniform in time convergence for the graphon particle system.Stochastic Processes and their Applications, 150:532–568, 2022

    2022

  3. [9]

    Graphon particle system: Uniform-in-time concentration bounds.Stochastic Processes and their Applications, 156:196–225, 2023

    Erhan Bayraktar and Ruoyu Wu. Graphon particle system: Uniform-in-time concentration bounds.Stochastic Processes and their Applications, 156:196–225, 2023

    2023

  4. [10]

    Propagation of chaos of forward–backward stochastic differential equations with graphon interactions.Applied Mathematics & Optimization, 88(1):25, 2023

    Erhan Bayraktar, Ruoyu Wu, and Xin Zhang. Propagation of chaos of forward–backward stochastic differential equations with graphon interactions.Applied Mathematics & Optimization, 88(1):25, 2023. 21

    2023

  5. [11]

    Non-parametric estimates for graphon mean-field particle systems.Bernoulli, 31(4):2940– 2961, 2025

    Erhan Bayraktar and Hongyi Zhou. Non-parametric estimates for graphon mean-field particle systems.Bernoulli, 31(4):2940– 2961, 2025

    2025

  6. [12]

    Springer, 2013

    Alain Bensoussan, Jens Frehse, and Phillip Yam.Mean field games and mean field type control theory, volume 101. Springer, 2013

    2013

  7. [13]

    Springer, 2011

    Haim Brezis.Functional Analysis, Sobolev Spaces and Partial Differential Equations, volume 2. Springer, 2011

    2011

  8. [14]

    Path-dependent controlled mean-field coupled forward-backward sdes: The associated stochastic maximum principle.SIAM Journal on Control and Optimization, 63(3):2124–2153, 2025

    Rainer Buckdahn, Juan Li, Junsong Li, and Chuanzhi Xing. Path-dependent controlled mean-field coupled forward-backward sdes: The associated stochastic maximum principle.SIAM Journal on Control and Optimization, 63(3):2124–2153, 2025

    2025

  9. [15]

    Graphon mean field games and their equations.SIAM Journal on Control and Optimization, 59(6):4373–4399, 2021

    Peter E Caines and Minyi Huang. Graphon mean field games and their equations.SIAM Journal on Control and Optimization, 59(6):4373–4399, 2021

    2021

  10. [16]

    Probabilistic analysis of graphon mean field control.arXiv preprint arXiv:2505.19664, 2025

    Zhongyuan Cao and Mathieu Laurière. Probabilistic analysis of graphon mean field control.arXiv preprint arXiv:2505.19664, 2025

    2025

  11. [17]

    Springer, Cham, 2018

    René Carmona and François Delarue.Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games, volume 83 ofProbability Theory and Stochastic Modelling. Springer, Cham, 2018

    2018

  12. [18]

    Springer, Cham, 2018

    René Carmona and François Delarue.Probabilistic Theory of Mean Field Games with Applications II: Mean Field Games with Common Noise and Master Equations, volume 84 ofProbability Theory and Stochastic Modelling. Springer, Cham, 2018

    2018

  13. [19]

    Mean fieldgamesandsystemicrisk.Communications in Mathematical Sciences, 13(4):911–933, 2015

    RenéA.Carmona, Jean-PierreFouque, andLi-HsienSun. Mean fieldgamesandsystemicrisk.Communications in Mathematical Sciences, 13(4):911–933, 2015

    2015

  14. [20]

    Dissipative backward stochastic differential equations with locally lipschitz nonlinearity.Stochastic Processes and their Applications, 117(5):613–628, 2007

    Fulvia Confortola. Dissipative backward stochastic differential equations with locally lipschitz nonlinearity.Stochastic Processes and their Applications, 117(5):613–628, 2007

    2007

  15. [21]

    Nonlinear graphon mean-field systems.Stochastic Processes and their Applications, page 104728, 2025

    Fabio Coppini, Anna De Crescenzo, and Huyên Pham. Nonlinear graphon mean-field systems.Stochastic Processes and their Applications, page 104728, 2025

    2025

  16. [22]

    Optimal control of path-dependent mckean–vlasov sdes in infinite dimension.Annals of Applied Probability, 33(4):2863–2918, 2023

    Andrea Cosso, Fausto Gozzi, Idris Kharroubi, Huyên Pham, and Mauro Rosestolato. Optimal control of path-dependent mckean–vlasov sdes in infinite dimension.Annals of Applied Probability, 33(4):2863–2918, 2023

    2023

  17. [23]

    Mean field optimal stopping with uncontrolled state.arXiv preprint arXiv:2503.04269, 2025

    Andrea Cosso and Laura Perelli. Mean field optimal stopping with uncontrolled state.arXiv preprint arXiv:2503.04269, 2025

    2025 arXiv

  18. [25]

    Mean-field control of non exchangeable systems

    Anna De Crescenzo, Marco Fuhrman, Idris Kharroubi, and Huyên Pham. Mean-field control of non exchangeable systems. arXiv preprint arXiv:2407.18635, 2024

    2024 arXiv

  19. [26]

    Sensitivity of functionals of mckean–vlasov sde’s with respect to the initial distribution.arXiv preprint arXiv:2412.15906, 2024

    Filippo de Feo, Salvatore Federico, Fausto Gozzi, and Nizar Touzi. Sensitivity of functionals of mckean–vlasov sde’s with respect to the initial distribution.arXiv preprint arXiv:2412.15906, 2024

    2024 arXiv

  20. [27]

    Stochastic optimal control of interacting particle systems in hilbert spaces and applications.arXiv preprint, 2025

    Filippo de Feo, Fausto Gozzi, Andrzej Święch, and Lukas Wessels. Stochastic optimal control of interacting particle systems in hilbert spaces and applications.arXiv preprint, 2025. To appear

    2025

  21. [28]

    Optimal portfolio choice with path dependent benchmarked labor income: a mean field model.Stochastic Processes and their Applications, 145:48–85, 2022

    Boualem Djehiche, Fausto Gozzi, Giovanni Zanco, and Margherita Zanella. Optimal portfolio choice with path dependent benchmarked labor income: a mean field model.Stochastic Processes and their Applications, 145:48–85, 2022

    2022

  22. [29]

    A non–exchangeable mean field control problem with controlled interactions.arXiv preprint arXiv:2511.00288, 2025

    Mao Fabrice Djete. A non–exchangeable mean field control problem with controlled interactions.arXiv preprint arXiv:2511.00288, 2025

    2025

  23. [30]

    Mean field game of mutual holding with defaultable agents, and systemic risk.SIAM Journal on Control and Optimization, 62(2):1234–1278, 2024

    Mao Fabrice Djete, Gaoyue Guo, and Nizar Touzi. Mean field game of mutual holding with defaultable agents, and systemic risk.SIAM Journal on Control and Optimization, 62(2):1234–1278, 2024

    2024

  24. [31]

    Mean field game of mutual holding.The Annals of Applied Probability, 34(6):4999–5031, 2024

    Mao Fabrice Djete and Nizar Touzi. Mean field game of mutual holding.The Annals of Applied Probability, 34(6):4999–5031, 2024

    2024

  25. [32]

    Stochastic control for mean-field stochastic partial differential equations with jumps.Journal of Optimization Theory and Applications, 176(3):559–584, 2018

    Roxana Dumitrescu, Bernt Øksendal, and Agnès Sulem. Stochastic control for mean-field stochastic partial differential equations with jumps.Journal of Optimization Theory and Applications, 176(3):559–584, 2018

    2018

  26. [33]

    Springer, 2017

    Giorgio Fabbri, Fausto Gozzi, and Andrzej Święch.Stochastic Optimal Control in Infinite Dimension. Springer, 2017

    2017

  27. [34]

    Linear-quadratic mean field games in hilbert spaces.SIAM Journal on Mathematical Analysis, 57(6):5821–5853, 2025

    Salvatore Federico, Daria Ghilli, and Fausto Gozzi. Linear-quadratic mean field games in hilbert spaces.SIAM Journal on Mathematical Analysis, 57(6):5821–5853, 2025

    2025

  28. [35]

    On mean field games in infinite dimension.Journal de Mathématiques Pures et Appliquées, page 103780, 2025

    Salvatore Federico, Fausto Gozzi, and Andrzej Święch. On mean field games in infinite dimension.Journal de Mathématiques Pures et Appliquées, page 103780, 2025

    2025

  29. [36]

    A unified approach to linear-quadratic-gaussian mean-field team: homogeneity, heterogeneity and quasi-exchangeability.The Annals of Applied Probability, 33(4):2786–2823, 2023

    Xinwei Feng, Ying Hu, and Jianhui Huang. A unified approach to linear-quadratic-gaussian mean-field team: homogeneity, heterogeneity and quasi-exchangeability.The Annals of Applied Probability, 33(4):2786–2823, 2023

    2023

  30. [37]

    Simultaneously solving infinitely many lq mean field games in hilbert spaces: The power of neural operators.arXiv preprint arXiv:2510.20017, 2025

    Dena Firoozi, Anastasis Kratsios, and Xuwei Yang. Simultaneously solving infinitely many lq mean field games in hilbert spaces: The power of neural operators.arXiv preprint arXiv:2510.20017, 2025. 22

    2025

  31. [38]

    Hilbert space-valued lq mean field games: An infinite-dimensional analysis.SIAM Journal on Control and Optimization, 63(5):3297–3327, 2025

    Dena Firoozi and Hanchao Liu. Hilbert space-valued lq mean field games: An infinite-dimensional analysis.SIAM Journal on Control and Optimization, 63(5):3297–3327, 2025

    2025

  32. [39]

    Mean field game with delay: A toy model.Risks, 6(3):1–19, 2018

    Jean-Pierre Fouque and Zhaoyu Zhang. Mean field game with delay: A toy model.Risks, 6(3):1–19, 2018

    2018

  33. [40]

    Actor–critic learning for mean-field control in continuous time.Journal of Machine Learning Research, 26:1–42, 2025

    Noufel Frikha, Maxence Germain, Mathieu Laurière, Huyên Pham, and Xuanye Song. Actor–critic learning for mean-field control in continuous time.Journal of Machine Learning Research, 26:1–42, 2025

    2025

  34. [41]

    An optimal advertising model with carryover effect and mean field terms.Mathematics and Financial Economics, 18(2–3):413–427, 2024

    Fausto Gozzi, Federica Masiero, and Mauro Rosestolato. An optimal advertising model with carryover effect and mean field terms.Mathematics and Financial Economics, 18(2–3):413–427, 2024

    2024

  35. [42]

    On the backward stochastic riccati equation in infinite dimensions.SIAM Journal on Control and Optimization, 44(1):159–194, 2005

    Giuseppina Guatteri and Gianmario Tessitore. On the backward stochastic riccati equation in infinite dimensions.SIAM Journal on Control and Optimization, 44(1):159–194, 2005

    2005

  36. [43]

    A linear–quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon.Mathematical Control and Related Fields, 5(1):97–139, 2015

    Jianhui Huang, Xun Li, and Jiongmin Yong. A linear–quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon.Mathematical Control and Related Fields, 5(1):97–139, 2015

    2015

  37. [44]

    Mean-field limit of non-exchangeable systems.Communications on Pure and Applied Mathematics, 78(4):651–741, 2025

    Pierre-Emmanuel Jabin, David Poyato, and Juan Soler. Mean-field limit of non-exchangeable systems.Communications on Pure and Applied Mathematics, 78(4):651–741, 2025

    2025

  38. [46]

    A label-state formulation of stochastic graphon games and approximate equilibria on large networks.Mathematics of Operations Research, 48(4):1987–2018, 2023

    Daniel Lacker and Agathe Soret. A label-state formulation of stochastic graphon games and approximate equilibria on large networks.Mathematics of Operations Research, 48(4):1987–2018, 2023

    1987

  39. [47]

    Mean field games.Japanese Journal of Mathematics, 2(1):229–260, 2007

    Jean-Michel Lasry and Pierre-Louis Lions. Mean field games.Japanese Journal of Mathematics, 2(1):229–260, 2007

    2007

  40. [48]

    An overview of some extensions of mean field games beyond perfect homogeneity and anonymity.arXiv preprint arXiv:2511.04929, 2025

    Mathieu Laurière. An overview of some extensions of mean field games beyond perfect homogeneity and anonymity.arXiv preprint arXiv:2511.04929, 2025

    2025

  41. [49]

    American Mathematical Society, 2010

    László Lovász.Large Networks and Graph Limits. American Mathematical Society, 2010

    2010

  42. [50]

    Analysis of non-exchangeable mean field markov decision processes with common noise: From bellman equation to quantitative propagation of chaos, 2025

    Samy Mekkaoui and Huyên Pham. Analysis of non-exchangeable mean field markov decision processes with common noise: From bellman equation to quantitative propagation of chaos, 2025. Work in progress

    2025

  43. [51]

    Stochastic graphon games with memory.arXiv preprint arXiv:2411.05896, 2024

    Eyal Neuman and Sturmius Tuschmann. Stochastic graphon games with memory.arXiv preprint arXiv:2411.05896, 2024

    2024 arXiv

  44. [52]

    Linearquadraticoptimalcontrolofconditionalmckean–vlasovequationwithrandomcoefficientsandapplications

    HuyênPham. Linearquadraticoptimalcontrolofconditionalmckean–vlasovequationwithrandomcoefficientsandapplications. Probability, Uncertainty and Quantitative Risk, 1:7, 2016

    2016

  45. [53]

    Dynamic programming for optimal control of stochastic mckean–vlasov dynamics.SIAM Journal on Control and Optimization, 55(2):1069–1101, 2017

    Huyên Pham and Xiaoli Wei. Dynamic programming for optimal control of stochastic mckean–vlasov dynamics.SIAM Journal on Control and Optimization, 55(2):1069–1101, 2017

    2017

  46. [54]

    Entropic optimal planning for path-dependent mean field games

    Zhenjie Ren, Xiaolu Tan, Nizar Touzi, and Junjian Yang. Entropic optimal planning for path-dependent mean field games. SIAM Journal on Control and Optimization, 61(3):1415–1437, 2023

    2023

  47. [55]

    Mean field games incorporating carryover effects: Optimizing advertising models

    Michele Ricciardi and Mauro Rosestolato. Mean field games incorporating carryover effects: Optimizing advertising models. Decisions in Economics and Finance, 2024

    2024

  48. [56]

    L. C. G. Rogers and David Williams.Diffusions, Markov Processes and Martingales, Volume 2: Itô Calculus. Cambridge University Press, 2000

    2000

  49. [57]

    Mean-field backward stochastic volterra integral equations.Discrete and Continuous Dynamical Systems Series B, 18(7):1929–1967, 2013

    Yufeng Shi, Tianxiao Wang, and Jiongmin Yong. Mean-field backward stochastic volterra integral equations.Discrete and Continuous Dynamical Systems Series B, 18(7):1929–1967, 2013

    1929

  50. [58]

    Pontryagin maximum principle for mckean–vlasov stochastic reaction–diffusion equations.arXiv preprint arXiv:2507.16288, 2025

    Johannes Benedikt Spille and Wilhelm Stannat. Pontryagin maximum principle for mckean–vlasov stochastic reaction–diffusion equations.arXiv preprint arXiv:2507.16288, 2025

    2025 arXiv

  51. [59]

    Mean-field stochastic linear quadratic optimal control problems: Open-loop solvabilities.ESAIM: Control, Optimization and Calculus of Variations, 23(3):1099–1127, 2017

    Jingrui Sun. Mean-field stochastic linear quadratic optimal control problems: Open-loop solvabilities.ESAIM: Control, Optimization and Calculus of Variations, 23(3):1099–1127, 2017

    2017

  52. [60]

    The exact law of large numbers via fubini extension and characterization of insurable risks.Journal of Economic Theory, 126(1):31–69, 2006

    Yeneng Sun. The exact law of large numbers via fubini extension and characterization of insurable risks.Journal of Economic Theory, 126(1):31–69, 2006

    2006

  53. [61]

    Dynamic programming equation for the mean field optimal stopping problem

    Mehdi Talbi, Nizar Touzi, and Jianfeng Zhang. Dynamic programming equation for the mean field optimal stopping problem. SIAM Journal on Control and Optimization, 61(4):2140–2164, 2023

    2023

  54. [62]

    Viscosity solutions for obstacle problems on wasserstein space.SIAM Journal on Control and Optimization, 61(3):1712–1736, 2023

    Mehdi Talbi, Nizar Touzi, and Jianfeng Zhang. Viscosity solutions for obstacle problems on wasserstein space.SIAM Journal on Control and Optimization, 61(3):1712–1736, 2023

    2023

  55. [63]

    From finite population optimal stopping to mean field optimal stopping.Annals of Applied Probability, 34(5):4237–4267, 2024

    Mehdi Talbi, Nizar Touzi, and Jianfeng Zhang. From finite population optimal stopping to mean field optimal stopping.Annals of Applied Probability, 34(5):4237–4267, 2024

    2024

  56. [64]

    Forward and backward mean-field stochastic partial differential equation and optimal control.Chinese Annals of Mathematics, Series B, 40(4):515–540, 2019

    Maoning Tang, Qingxin Meng, and Meijiao Wang. Forward and backward mean-field stochastic partial differential equation and optimal control.Chinese Annals of Mathematics, Series B, 40(4):515–540, 2019. 23

    2019

  57. [65]

    Linear-quadratic graphon mean field games with common noise

    De-xuan Xu, Zhun Gou, Nan-jing Huang, and Shuang Gao. Linear-quadratic graphon mean field games with common noise. SIAM Journal on Control and Optimization, 63(5):3526–3556, 2025

    2025

  58. [66]

    Linear-quadratic optimal control problems for mean-field stochastic differential equations.SIAM Journal on Control and Optimization, 51(4):2809–2838, 2013

    Jiongmin Yong. Linear-quadratic optimal control problems for mean-field stochastic differential equations.SIAM Journal on Control and Optimization, 51(4):2809–2838, 2013. 24

    2013

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.