Mean Field Game for Linear Quadratic Stochastic Recursive Systems

Liangquan Zhang; Xun Li

REVIEW 3 major objections 4 minor 47 references

Mean Field Game for Linear Quadratic Stochastic Recursive Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that a large population of agents with forward-backward state dynamics and convex control constraints can be coordinated by projection-based decentralized strategies that form an ε-Nash equilibrium with ε = O(1/√N).

desk verdict A useful extension of constrained LQ mean-field games to FBSDE dynamics, but the main epsilon-Nash theorem is only proved for decentralized deviations while stated for centralized ones, and an undefined symbol in the well-posedness proof needs fixing. read the letter →

arxiv 1908.05063 v1 pith:BDZEHWNV submitted 2019-08-14 math.OC

Liangquan Zhang , Xun Li This is my paper · ORCID

classification math.OC MSC 93E2060H1560H30

keywords mean-fieldgameslinear-quadraticcontrolforward-backwardstochasticdifferentialequationsconvexconstraintsprojectionoperatorepsilon-Nashequilibriumconsistencyconditionmonotonicity

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

Large populations of agents whose states evolve through forward-backward stochastic differential equations, each with a convex constraint on its control, are usually impossible to coordinate centrally. This paper tries to show they can be coordinated almost optimally by a simple decentralized rule: each agent solves its own linear-quadratic problem with the population averages frozen, and the resulting consistency condition becomes a coupled mean-field forward-backward SDE with a projection operator. The paper proves that this consistency system has a unique solution under a monotonicity condition, and that the decentralized strategies form an ε-Nash equilibrium with ε = O(1/√N). This matters because recursive utility and terminal-benchmark problems, such as fund-manager performance evaluation, have exactly this forward-backward structure.

What carries the argument

The load-bearing object is the consistency-condition system (10), a coupled mean-field forward-backward SDE in the variables (x,y,z,p,q,k). Its nonlinearity comes from the projection operator ϕ(p,q,k)=PU[$R^{{-1}}$(B^T q + K^T p + D^T k)], which encodes the closed convex control constraint, for example the no-shorting constraint U = R^m_+. The projection is monotone and Lipschitz, and these two properties drive both the well-posedness proof, via a continuation method, and the ε-Nash estimates.

What would settle it

Construct a numerical instance satisfying (A1)-(A2), say with U = R_+ and N large, solve the consistency-condition system, and search over controls in the centralized admissible set that use the full population Brownian motions; if any such control beats the claimed O(1/√N) bound uniformly in N, Theorem 5 as stated fails. Short of that, an analytic counterexample to the private-filtration restriction would settle the question.

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Extended reading notes

Core claim

The central claim is Theorem 5: under assumptions (A1)-(A2), the decentralized strategies (u̅^1,...,u̅^N), each equal to the projection ϕ(χ^i,β^i,γ^i) = PU[$R^{{-1}}$(B^T β^i + K^T χ^i + D^T γ^i)], form an ε-Nash equilibrium of the N-agent problem, with ε = O(1/√N). In other words, each agent's cost when everyone follows the rule is within O(1/√N) of its cost under any other admissible strategy, as admissibility is defined in the paper. The argument combines the stochastic maximum principle for convex control sets, mean-field law of large numbers, and a continuation method for the coupled consistency FBSDE.

Load-bearing premise

The ε-Nash proof only checks deviations that use the deviating agent's own private information, while the equilibrium definition allows deviations that see the whole population's noise; the paper does not show this gap is harmless.

Editorial extensions

If this is right

The consistency-condition system can be solved off line, so each agent's strategy depends only on its own noise and the frozen means, not on the full population.
When the control set is the positive orthant, the projection acts as a no-shorting constraint, so the same framework covers constrained portfolio and hedging problems with recursive costs.
The O(1/√N) rate means the per-agent suboptimality vanishes as the population grows, giving an explicit bound on the Nash gap for finite N.
The well-posedness result characterizes the mean-field limit of forward-backward games, extending the usual LQ mean-field game from forward states to recursive systems.

Reading between the lines

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

Beyond the paper, if the restriction to private-filtration deviations is only technical, the same projection-based argument may extend to closed-loop deviations, because the projection is Lipschitz with respect to state feedback; the paper does not test this.
The consistency-condition FBSDE suggests a numerical route: discretize and solve the monotone FBSDE once, then use the projection as a lookup feedback law; the paper gives no algorithm, but the monotonicity would make a fixed-point iteration natural.
A second-order refinement could replace the frozen mean Eα with the empirical average including fluctuations, potentially improving the rate beyond O(1/√N); the paper stops at the first-order consistency condition.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, the paper extends the Hu-Huang-Li [24] program for constrained LQ mean-field games to FBSDE dynamics with convex control constraints, and the consistency condition becomes a coupled MF-FBSDE with projection operators. That combination is new and worth knowing about. Second, the main theorem as stated is not proved: the epsilon-Nash statement in Definition 3 and Problem (CC) quantifies over centralized controls adapted to the full N-agent filtration, but the proof of Theorem 5 only treats decentralized deviations adapted to the deviating agent's own Brownian motion. The final chain uses the optimality of the candidate strategy in the limiting problem, which was established only over decentralized controls. So the theorem is only established for a weaker, decentralized version of the equilibrium.

What the paper does well: the projection-based maximum principle step is standard and applied correctly; the estimates in Lemmas 7-12 are the expected law-of-large-numbers fluctuation arguments and look plausible; the monotonicity/continuation framework for well-posedness follows the Hu-Peng method and is a reasonable route. The fund-management motivation with no-shorting constraints is legitimate.

Soft spots: an undefined symbol Psi appears twice — in the initial condition of system (13) and in the uniqueness proof of Theorem 2 — and neither occurrence makes sense. The initial condition for chi is also inconsistent: 0 in (10)-(11), but -Psi(theta0 - E theta0) in (13). The contraction step in Theorem 2 is sketched and leans on [20]; that is acceptable in this literature, but a referee would want the details. The quantifier mismatch is the load-bearing issue. It might be fixable by redefining the equilibrium over decentralized controls, or by proving that centralized deviations cannot do better than decentralized ones for the limiting cost, but neither fix is present.

Who this is for: researchers in mean-field games and stochastic control working on constrained LQ problems. A reading group could get value from the centralized-versus-decentralized admissible-control subtlety, though not from the typos.

Recommendation: I would send it to peer review. The combination is new, the approach is coherent, and the flaws look repairable. I would not cite it in its current form.

Referee Report

3 major / 4 minor

Summary. The paper studies linear-quadratic mean-field games in which each agent's state evolves by a forward-backward SDE with a convex control constraint. The authors derive a consistency-condition system that is a coupled mean-field FBSDE with a projection operator, prove its well-posedness by a monotonicity/continuation argument, and then claim that the resulting decentralized open-loop strategies form an ε-Nash equilibrium for the finite-N population problem with ε = O(1/√N). The main technical steps mirror the method of Hu–Huang–Li [24]: a stochastic maximum principle for the limiting control problem, a fixed-point/consistency construction, and perturbation estimates that compare the finite-N and limiting states.

Significance. If fully established, the result would extend mean-field LQG theory to forward-backward dynamics with control constraints, a setting relevant to recursive utility and performance-evaluation applications. The paper provides a plausible route: the consistency-condition system with a projection is a natural generalization of [24] to FBSDE dynamics, and the O(1/√N) rate is the expected one. However, the two load-bearing gaps discussed below mean that the central claim is not proven as stated. The paper also does not supply machine-checked proofs or reproducible code, so the assessment rests entirely on the written derivation.

major comments (3)

[Section 3, Theorem 5, proof before Eq. (25)] Theorem 5, as stated via Definition 3 and Problem (CC), requires the ε-Nash inequality for all alternative strategies u^i in the centralized set U^c_ad, i.e., controls adapted to the full population filtration F. However, the proof of Theorem 5 explicitly restricts the perturbation to u^i ∈ U^{d,i}_ad, adapted only to the agent's own Brownian filtration. The final chain J_i(ū^i, ū^{-i}) = J̄_i(ū^i) + O(N^{-1/2}) ≤ J̄_i(u^i) + O(N^{-1/2}) = J_i(u^i, ū^{-i}) + O(N^{-1/2}) relies on J̄_i(ū^i) ≤ J̄_i(u^i), which was proved in Section 2 only for u^i ∈ U^{d,i}_ad via the maximum principle. The paper gives no argument that the infimum of the limiting cost over U^c_ad coincides with the infimum over U^{d,i}_ad, or that a centralized deviation cannot exploit the other agents' noises. This is a quantifier mismatch between the statement and the proof; the theorem is therefore not established as written. A repair would either weaken Definition 3 and Problem (CC) to decentralized deviations or add a lemma showing the two infima coincide.
[Appendix A, Proof of Theorem 2 (uniqueness)] The uniqueness proof of Theorem 2 contains an undefined symbol Ψ. After applying Itô's formula to ⟨q̂, x̂⟩ − ⟨p̂, ŷ⟩ and taking expectations, the paper writes "0 = E[⟨G(x̂_T − Ex̂_T), x̂_T⟩ + Ψŷ_0(ŷ_0 − Eŷ_0)] + ..." with no prior definition of Ψ. The symbol also appears in system (13) as the initial condition χ^i_0 = −Ψ(θ^i_0 − Eθ^i_0), whereas the consistency-condition system (11) has χ^i_0 = 0. Since the subsequent inequality and the conclusion of the uniqueness argument depend on the sign and form of the boundary terms, the undefined object makes the proof incomplete. Moreover, the displayed equality appears to have a sign inconsistency: with q_T = Φ^T p_T − G(x_T − Ex_T), the boundary contribution ⟨q_T, x_T⟩ − ⟨p_T, y_T⟩ equals −⟨G(x_T − Ex_T), x_T⟩, whereas the proof uses a positive sign before passing to the lower bound.
[Appendix A, Lemma 13 and proof of existence] The continuation argument for existence of solutions to the consistency-condition system is not completed. In Lemma 13 the mapping I_{α0+δ0} is introduced, and estimates (39)–(43) are displayed, but the decisive step is only announced: "Combining (39)-(43), by similar method used in [20], we have ... a contraction." The norm in which the contraction is asserted is not fully specified, the roles of the constants C1,...,C5 and the choice of δ0 are not given, and the cancellation of the (x̂^{i+1}, ŷ^{i+1}) terms from the left-hand side is not shown. Since the well-posedness of the consistency-condition system (10) underlies the construction of the decentralized strategies, this gap directly affects the main theorem. The proof needs to be written out or replaced with a precise reference that states the exact estimates being invoked.

minor comments (4)

[Section 2, system (10)] The fourth equation of system (10) reads "dq = [−Mp − Aq + Q(x − Ex)]dt + kWt", which should presumably be "k dW_t"; the same typo appears in system (11).
[Section 3, system (13)] In system (13) the initial condition for χ^i is written χ^i_0 = −Ψ(θ^i_0 − Eθ^i_0), but system (11) has χ^i_0 = 0. This inconsistency is confusing because both systems are said to describe the same consistency-condition solution; please align the notation.
[Appendix A, equation (35)] In the parametrized system (35) the term "γ − Eγ" appears although γ is not defined in the list of inputs; presumably it should be γ0. Similarly, "µ0" appears in the text but is not in the stated input tuple (b0, σ0, γ0, λ0, µ0, ψ0); please correct the notation.
[Throughout] There are frequent typos and OCR artifacts, e.g., "Φ^T T" in the terminal conditions of (11) and (13), "obatin" in the proof of Lemma 11, and "baisc technique" in Lemma 13. A careful proofreading pass would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the consistency-condition system and ε-Nash estimates are derived in-paper; the only noted gap (deviations restricted to U^{d,i}_ad) is a correctness issue, not a circular reduction.

full rationale

The paper does not fit a parameter and rename it a prediction, nor does it define its target in terms of its inputs. The MFG strategy is characterised by the projection formula (7) obtained from the maximum-principle condition (6), and the consistency-condition system (10) is solved for (x,y,z,p,q,k) by a self-contained continuation argument in Theorem 2. The ε-Nash proof expands the cost differences through Lemmas 7–12; each estimate is proved with explicit O(1/N) or O(1/sqrt(N)) bounds rather than imported. The main self-reference is to Hu-Huang-Li [24], which shares author X. Li and supplies the general strategy of using the stochastic maximum principle with a convex control constraint, but the paper explicitly develops its own mean-field FBSDE with projection and proves the needed well-posedness. The strongest concern raised by a skeptical reader is that Definition 3 and Problem (CC) quantify the ε-Nash inequality over centralized controls U^c_ad, while the proof of Theorem 5 only considers perturbations ui in U^{d,i}_ad and uses optimality of the limiting control in Problem (LCC), which is established only for decentralized controls. That is a genuine gap between theorem statement and proof, but it is not circularity: the theorem does not reduce by construction to its assumptions, and no fitted or self-referential input forces the conclusion. Under the supplied rubric, this is a normal non-circular finding.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard convex analysis, the stochastic maximum principle, and a continuation argument for mean-field FBSDEs. There are no free parameters fitted to data and no invented physical entities. The main unstated load is the contraction estimate in the existence proof and the unproved sufficiency of decentralized deviations in the ε-Nash proof.

assumptions (4)

standard math Closed convex control set U and the projection PU satisfy strict monotonicity and nonexpansiveness (Propositions 15-17, Appendix B).
Used to derive the form of the optimal decentralized control ui,* = PU[R^{-1}(B^T q + K^T p + D^T k)] and in the uniqueness proof of Theorem 2.
domain assumption The stochastic maximum principle for fully coupled forward-backward SDEs (Theorem 3.3 of Wu [43]) is assumed to hold for the auxiliary control problem (LCC).
Invoked in Section 2 to derive the adjoint system and the projection form of the optimal control.
standard math The law of large numbers identifies the population averages with expectations Ex and Ey in equations (8)-(9).
Basis for freezing φ1 and φ2 and for the mean-field approximation.
ad hoc to paper The continuation method for the coupled MF-FBSDE (10) is assumed to yield a contraction for some δ0; the combination of estimates (39)-(43) is only referenced to Hu-Peng [20].
Existence of a unique solution to the consistency-condition system depends on this step, which is not fully proved in the paper.

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Pith. "Pith review of Mean Field Game for Linear Quadratic Stochastic Recursive Systems." pith.science (2026). https://pith.science/paper/BDZEHWNV

@misc{pith2026190805063,
  author       = {Pith},
  title        = {Pith review of: Mean Field Game for Linear Quadratic Stochastic Recursive Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDZEHWNV}},
  note         = {Machine review of arXiv:1908.05063}
}

abstract

This paper focuses on linear-quadratic (LQ for short) mean-field games described by forward-backward stochastic differential equations (FBSDEs for short), in which the individual control region is postulated to be convex. The decentralized strategies and consistency condition are represented by a kind of coupled mean-field FBSDEs with projection operators. The well-posedness of consistency condition system is obtained using the monotonicity condition method. The $\epsilon$-Nash equilibrium property is discussed as well.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

47 extracted references · 46 canonical work pages

[24]

Y. Hu, J. Huang, X. Li. Linear quadratic mean ﬁeld game wi th control input con- straint. ESAIM COCV, 24 (2), 901-919, 2018

work page 2018
[20]

Hu and S

Y. Hu and S. Peng. Solutions of forward-backward stocha stic diﬀerential equations. Probability Theory and Related Fields , 103, 273-283 (1995)

work page 1995
[1]

Andersson and B

D. Andersson and B. Djehiche, A maximum principle for SDEs of meanﬁeld type, Appl. Math. Optim ., vol. 63, 341–356, 2011

work page 2011
[2]

An introductory approach to duality in optima l stochastic control,

J. Bismut, “An introductory approach to duality in optima l stochastic control,” SIAM Rev., vol. 20, 62–78, 1978

work page 1978
[3]

Bardi, Explicit solutions of some linear-quadratic me an ﬁeld games, Netw

M. Bardi, Explicit solutions of some linear-quadratic me an ﬁeld games, Netw. Heterog. Media, vol. 7, pp. 243–261, 2012

work page 2012
[4]

Buckdahn, B

R. Buckdahn, B. Djehiche, and J. Li, A general stochastic ma ximum principle for SDEs of mean-ﬁeld type, Appl. Math. Optim ., vol. 64, 197–216, 2011

work page 2011
[5]

Buckdahn, B

R. Buckdahn, B. Djehiche, J. Li, S.Peng, Mean-ﬁeld backwar d stochastic diﬀerential equations. A limit approach. Ann. Probab. 37 (4), 1524-1565, (2009)

work page 2009
[6]

Buckdahn, J

R. Buckdahn, J. Li, S. Peng, Mean-ﬁeld backward stochasti c diﬀerential equations and related partial diﬀerential equations. Stoch. Process. Appl . 119 (10), 3133-3154, (2009)

work page 2009

Show all 47 references

[7]

Balakrishnan

A.V. Balakrishnan. Applied Functional Analysis . Springer-Verlag, New York, 1976

1976
[8]

Bensoussan, J

A. Bensoussan, J. Frehse and S. Yam. Mean Field Games and Mean Field Type Control Theory. Springer, New York, 2013. 27

2013
[9]

H. Brezis. Functional Analysis, Sobolev Spaces and Partial Diﬀerenti al Equations . Springer, New York, 2011

2011
[10]

Bensoussan, K

A. Bensoussan, K. Sung, S. Yam, and S. Yung, Linear-quadr atic mean-ﬁeld games, preprint, 2014

2014
[11]

Carmona and F

R. Carmona and F. Delarue. Probabilistic analysis of me an-ﬁeld games. SIAM J.Control Optim., 51, 2705-2734 (2013)

2013
[12]

Carmona, J.P

R. Carmona, J.P. Fouque and L.H. Sun. Mean ﬁeld games and systemic risk. Com- munications in Mathematical Sciences , 13, 911-933 (2015)

2015
[13]

Cardaliaguet

P. Cardaliaguet. Notes on Mean Field Games, (2012)

2012
[14]

Duﬃe and L

D. Duﬃe and L. Epstein, Stochastic diﬀerential utility , Econometrica, vol. 60, 353– 394, 1992

1992
[15]

K. Du, J. Huang, and Z. Wu . Linear quadratic mean-ﬁeld-g ame of backward stochastic diﬀerential systems. Mathematical Control Related Fields , 8.3&4: 653-678, (2018)

2018
[16]

K. Du, J. Huang, and Z. Wu, Relationship between backwar d and forward linear- quadratic mean-ﬁeld-game with terminal constraint and opt imal asset allocation for insurers and pension funds. International Journal of Control , (2019)

2019
[17]

Espinosa and N

G.E. Espinosa and N. Touzi. Optimal investment under re lative performance concerns. Mathematical Finance, 25, 221-257 (2015)

2015
[18]

Guéant, J.-M

O. Guéant, J.-M. Lasry, and P.-L. Lions, Mean ﬁeld games and applications, in Paris- Princeton Lectures on Mathematical Finance. Berlin, German y: Springer-Verlag, 2010

2010
[19]

Hu and X.Y

Y. Hu and X.Y. Zhou. Constrained stochastic LQ control w ith random coeﬃcients, and application to portfolio selection. SIAM J.Control Optim ., 44, 444-466 (2005)

2005
[21]

Huang, P.E

M. Huang, P.E. Caines and R. P. Malhamé. Distributed mul ti-agent decision-making with partial observations: Asymptotic Nash equilibria. Proceedings of the 17th Inter- national Symposium on Mathematical Theory of Networks and S ystems, Kyoto, Japan (2006)

2006
[22]

Huang and S

J. Huang and S. Wang. Dynamic optimization of large-pop ulation systems with partial information. Journal of Optimization Theory and Applications . Volume 168, Issue 1, 231-245, 2016

2016
[23]

Huang, S

J. Huang, S. Wang and Z. Wu. Backward-forward linear-qua dratic mean-ﬁeld games with major and minor agents. Probability, Uncertainty and Quantitative Risk , Decem- ber 2016, 1:8. 28

2016
[25]

Y. Hu, J. Huang, T. Nie. Linear-quadratic-gaussian mix ed mean-ﬁeld games with heterogeneous input constraints. SIAM J.Control Optim ., 56 4 (2018) 2835-2877

2018
[26]

Huang, S

J. Huang, S. Wang and Z. Wu. Backward mean-ﬁeld linear-qu adratic-gaussian (LQG) games: full and partial information. IEEE Transactions on Automatic Control Vol- ume: 61, Issue: 12, Dec. 2016

2016
[27]

Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM J

M. Huang, Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM J. Control Optim ., vol. 48, 3318–3353, 2010

2010
[28]

Huang, P

M. Huang, P. Caines, and R. Malhamé Large-population co st-coupled LQG problems with non-uniform agents: Individual-mass behavior and dec entralizedǫ-Nash equilib- ria, IEEE Trans. Autom. Control , vol. 52, 1560–1571, 2007

2007
[29]

Huang, P

M. Huang, P. Caines, and R. Malhamé Social optima in mean ﬁeld LQG control: Centralized and decentralized strategies, IEEE Trans. Autom. Control , vol. 57, 1736– 1751, 2012

2012
[30]

Huang, R

M. Huang, R. Malhamé and P. Caines, Large population sto chastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty e quivalence principle, Commun. Inform. Syst ., vol. 6, 221–251, 2006

2006
[31]

Kolokoltsov, M

V. Kolokoltsov, M. Troeva, and W. Yang, On the rate of con vergence for the mean- ﬁeld approximation of controlled diﬀusions with large numb er of players, Dynamic Games Appl ., doi: 10.1007/s13235-013-0095-6, Online First

doi
[32]

El Karoui, S

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

1997
[33]

Lim and X

E. Lim and X. Y. Zhou, Linear-quadratic control of backw ard stochastic diﬀerential equations, SIAM J. Control Optim ., vol. 40, no. 2, 450–474, 2001

2001
[34]

Li and J

T. Li and J. Zhang, Asymptotically optimal decentraliz ed control for large population stochastic multiagent systems, IEEE Trans. Autom. Control, vol. 53, 1643–1660, 2008

2008
[35]

Lasry and P.-L

J.-M. Lasry and P.-L. Lions, Mean ﬁeld games, Jpn. J. Math ., vol. 2, 229–260, 2007

2007
[36]

Nguyen and M

S. Nguyen and M. Huang, Linear-quadratic-Gaussian mix ed games with continuum- parametrized minor players, SIAM J. Control Optim ., vol. 50, 2907–2937, 2012

2012
[37]

Nourian and P

M. Nourian and P. Caines, ǫ-Nash mean ﬁeld game theory for nonlinear stochastic dynamical systems with major and minor agents, SIAM J.Control Optim ., vol. 51, pp. 3302–3331, 2013

2013
[38]

Pardoux and S

E. Pardoux and S. Peng, Adapted solution of backward sto chastic equation, Syst. Control Lett., vol. 14, 55–61, 1990. 29

1990
[39]

Pardoux and S

E. Pardoux and S. Peng, Backward doubly stochastic diﬀer ential equations and sys- tems of quasilinear SPDEs, Probab. Theory Relat. Fields , 98 (1994), 209-227

1994
[40]

Peng, Backward SDE and related g-expectation, Backwar d stochastic diﬀerential equations (N

S. Peng, Backward SDE and related g-expectation, Backwar d stochastic diﬀerential equations (N. El Karoui and L. Mazliak, eds.), Pitman Res. No tes Math. Ser., vol. 364, Longman, Harlow, (1997), pp. 141–159

1997
[41]

Peng and Z

S. Peng and Z. Wu. Fully coupled forward-backward stoch astic diﬀerential equations and applications to optimal control. SIAM J.Control Optim ., 37, 825-843 (1999)

1999
[42]

Tembine, Q

H. Tembine, Q. Zhu and T. Basar. Risk-sensitive mean-ﬁel d games. IEEE Trans. Automat. Control, 59, 835-850 (2014)

2014
[43]

Z. Wu. Maximum principle for optimal control problem of fully coupled forward- backward stochastic systems. Systems Science and Mathematical Science , Vol. 11 No.3 Jul. 1998

1998
[44]

Wang and J

B. Wang and J. Zhang. Mean ﬁeld games for large-populatio n multiagent systems with Markov jump parameters. SIAM J.Control Optim ., 50, 2308-2334 (2012)

2012
[45]

Yong and X.Y

J. Yong and X.Y. Zhou. Stochastic controls: Hamiltonian systems and HJB equation s. Springer-Verlag, New York, 1999

1999
[46]

Yong, Linear-quadratic optimal control problems fo r mean-ﬁeld stochastic diﬀeren- tial equations, SIAM J

J. Yong, Linear-quadratic optimal control problems fo r mean-ﬁeld stochastic diﬀeren- tial equations, SIAM J. Control Optim ., vol. 51, 2809–2838, 2013

2013
[47]

J. Yong. Linear-quadratic optimal control problem for mean-ﬁeld stochastic diﬀerential equations. SIAM J.Control Optim ., 51, 2809-2838 (2013). 30

2013

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [24]

Y. Hu, J. Huang, X. Li. Linear quadratic mean ﬁeld game wi th control input con- straint. ESAIM COCV, 24 (2), 901-919, 2018

work page 2018

[2] [20]

Hu and S

Y. Hu and S. Peng. Solutions of forward-backward stocha stic diﬀerential equations. Probability Theory and Related Fields , 103, 273-283 (1995)

work page 1995

[3] [1]

Andersson and B

D. Andersson and B. Djehiche, A maximum principle for SDEs of meanﬁeld type, Appl. Math. Optim ., vol. 63, 341–356, 2011

work page 2011

[4] [2]

An introductory approach to duality in optima l stochastic control,

J. Bismut, “An introductory approach to duality in optima l stochastic control,” SIAM Rev., vol. 20, 62–78, 1978

work page 1978

[5] [3]

Bardi, Explicit solutions of some linear-quadratic me an ﬁeld games, Netw

M. Bardi, Explicit solutions of some linear-quadratic me an ﬁeld games, Netw. Heterog. Media, vol. 7, pp. 243–261, 2012

work page 2012

[6] [4]

Buckdahn, B

R. Buckdahn, B. Djehiche, and J. Li, A general stochastic ma ximum principle for SDEs of mean-ﬁeld type, Appl. Math. Optim ., vol. 64, 197–216, 2011

work page 2011

[7] [5]

Buckdahn, B

R. Buckdahn, B. Djehiche, J. Li, S.Peng, Mean-ﬁeld backwar d stochastic diﬀerential equations. A limit approach. Ann. Probab. 37 (4), 1524-1565, (2009)

work page 2009

[8] [6]

Buckdahn, J

R. Buckdahn, J. Li, S. Peng, Mean-ﬁeld backward stochasti c diﬀerential equations and related partial diﬀerential equations. Stoch. Process. Appl . 119 (10), 3133-3154, (2009)

work page 2009