Regularity of Solutions of Mean-Field $G$-SDEs

Karl-Wilhelm Georg Bollweg; Thilo Meyer-Brandis

REVIEW 3 major objections 4 minor 34 references

Regularity of Solutions of Mean-Field $G$-SDEs

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Mean-field G-SDE solutions are Fréchet differentiable in random initial data, and the derivative is itself the unique solution of a linearized G-SDE.

desk verdict New differentiability calculus for mean-field G-SDEs, likely correct in substance, but the manuscript has enough typos and one unproved aggregation step that it needs a careful revision before I'd trust the statements as written. read the letter →

arxiv 2508.07867 v1 pith:VEAK64I7 submitted 2025-08-11 math.PR q-fin.MF

Karl-Wilhelm Georg Bollweg , Thilo Meyer-Brandis This is my paper · ORCID

classification math.PRq-fin.MF MSC 60H1060H30

keywords mean-fieldG-SDEG-BrownianmotionFréchetdifferentiabilityrandominitialconditionsublinearexpectationKnightianuncertaintysensitivityanalysisgames

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

The paper proves that the solution of a mean-field stochastic differential equation driven by G-Brownian motion is a smooth functional of its random initial condition. Under Lipschitz and Fréchet differentiability assumptions on the coefficients, the map sending the initial random vector to the solution process is continuously Fréchet differentiable, and its derivative is itself characterized as the unique solution of a linearized G-SDE. With stronger assumptions, the same holds for second order and mixed derivatives. This matters because it supplies the first sensitivity calculus for mean-field G-SDEs in the sublinear-expectation setting, giving tools needed for optimality conditions, gradient-type methods, and numerical differentiation of solutions under volatility uncertainty. A key structural step is the identification of the derivative as the sum of a direct term and a mean-field feedback correction.

What carries the argument

The load-bearing mechanism is the concatenation identity $X^{t,\xi,\xi} = X^{t,\xi}$ (Lemma 3.10), which uses the aggregation property of the conditional sublinear expectation to express the random-initial-condition problem as the deterministic-initial-condition equation evaluated at $x = \xi$. This identity reduces the Fréchet derivative of the mean-field solution map to two linearized G-SDEs: the derivative $D_x X^{t,\xi,\xi} \eta$ of the auxiliary process and the process $Y^{t,\xi,\eta}$ solving (4.4), which carries the law-feedback correction. The proof machinery consists of sublinear-expectation Grönwall estimates and the Burkholder–Davis–Gundy inequality in the G-framework (Lemma A.5),

What would settle it

Take the linear mean-field G-SDE with $b(s,x,\xi)=x+\xi$, $h=g=0$, and compute the closed-form solution; check directly whether $X^{t,\xi+\varepsilon\eta} - X^{t,\xi} - (D_x X^{t,\xi,\xi}\eta + Y^{t,\xi,\eta})$ tends to zero in $H^1_*$ at rate $o(\varepsilon)$. A concrete mismatch would falsify Proposition 4.23. Alternatively, exhibit an unbounded $\xi \in L^{2,d}_*(t)$ for which the aggregation identity $\|X^{t,\xi,\xi} - X^{t,\xi}\|_{H^2_*} = 0$ fails; since Lemma 3.10 underpins the whole derivative identification, such a counterexample would invalidate the main theorem.

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Extended reading notes

Core claim

The paper's main claim is that, for a mean-field G-SDE with coefficients satisfying Assumptions 3.1 and 4.3 and with $q_0 \ge 4$, the solution map $\xi \mapsto X^{t,\xi}$ from the space of square-integrable random initial vectors $L^{2,d}_*(t)$ to the process space $H^{1,d}_*(t,T)$ is continuously Fréchet differentiable (Proposition 4.23). The derivative at $\xi$ acting on a direction $\eta$ is explicit: $D_\xi X^{t,\xi} \eta = D_x X^{t,\xi,\xi} \eta + Y^{t,\xi,\eta}$. Here $D_x X^{t,\xi,\xi} \eta$ is the derivative of the solution with deterministic initial condition evaluated at the random point $x = \xi$, and $Y^{t,\xi,\eta}$ is the unique solution of the G-SDE (4.4), which encodes how th

Load-bearing premise

The argument assumes that the aggregation property of the conditional sublinear expectation remains valid on the completed square-integrable spaces $L^{2,d}_*$ and $H^{2,d}_*(t,T)$: that evaluating the deterministic-initial-condition solution process at $x = \xi$ actually recovers the mean-field solution quasi-surely, with the same identity for derivative processes. The paper invokes this from the G-framework background rather than proving it for the completed spaces; if it f

Editorial extensions

If this is right

The derivative formula provides a first-order sensitivity analysis for mean-field G-SDEs under volatility uncertainty, paralleling classical mean-field SDE results.
Under Assumption 5.1, the second and mixed Fréchet derivatives of the solution map are also unique solutions of G-SDEs, so Taylor expansions of the solution functional in the initial data become available.
These derivatives can be used to derive Pontryagin-type optimality conditions for mean-field control problems and to build (sub)gradient methods for optimizing over initial conditions in the G-setting.
The lifted derivative on $L^{2,d}_*$ suggests a Lions-type derivative on the space of sublinear distributions, as developed in Section 6, extending differentiability to coefficients that depend on the sublinear distribution.

Reading between the lines

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

The splitting $D_\xi X = D_x X + Y$ is likely to persist under weaker coefficient conditions than the boundedness assumed here, following the pattern of classical mean-field SDE theory, because the growth estimates in Section 3 only use Lipschitz and integrability bounds.
Since the derivative processes satisfy explicit G-SDEs, one can in principle simulate them jointly with the original solution, yielding a practical algorithm for differentiating robust worst-case expectations with respect to initial data.
If the aggregation property extends beyond the currently invoked background result, the same machinery should yield higher-order derivatives, giving a full Fréchet Taylor expansion of the mean-field solution map.
Connecting the derivative to the Lions derivative in the Wasserstein sense when the sublinear expectation is represented by a family of linear expectations would test whether the formula matches the classical mean-field derivative in the absence of uncertainty.

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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

The short version: this paper gives the first Fréchet-differentiability results for mean-field G-SDEs in the random-variable formulation of [2], and the main derivative formulas are plausible and new. I'd send it to a good referee, but I'd expect the referee to ask for a nontrivial number of corrections.

What is genuinely new: the map ξ ↦ X^{t,ξ} from L^2_*(t) to H^1_*(t,T) is shown continuously Fréchet differentiable, with derivative D_x X^{t,ξ,ξ} η + Y^{t,ξ,η}, where Y solves a linear G-SDE. For deterministic x, x↦X^{t,x,ξ} is C^1 into H^2. The proofs follow the standard Buckdahn et al. program and the Grönwall computations mostly check out. The reliance on [2] for well-posedness is appropriate.

The list of formal issues is longer than it should be. Lemma 4.2 states f(v0+v)-f(v) instead of f(v0+v)-f(v0). Proposition 4.23 claims a derivative into H^2 for a map into H^1 — presumably the intended target is H^1, and the proof actually works there. Lemma 3.8 has an exponent mismatch: the Grönwall term uses the square of the sup, not the p-th power. Assumption 5.1 uses κ(s) before defining it. And the abstract overstates the second-order scope: Section 5 gives D^2_x and D_xD_ξ, not a pure D^2_ξ. None of these are fatal, but they make the paper hard to read and should be fixed.

The one issue I'd want a referee to pin down is the aggregation/bootstrap used in Lemma 3.10 and again in Lemmas 4.12, 4.18, 4.22, and Proposition 4.23. The paper concludes, from a conditional estimate that holds for each deterministic x, that after substituting x=ξ the H^2 norm of X^{t,ξ,ξ}−X^{t,ξ} is zero. That substitution is standard for bounded continuous functions of the future in the G-setting, but here it's applied to square-integrable functionals on the completed spaces L^2_* and H^2_*, and no theorem is cited. Sublinear expectations are not continuous from below, so this is not a free step. I suspect it's true and can be proved by approximation with simple F-measurable ξ and dominated convergence; but as written it's a gap in the proof of the central result.

This paper is for specialists in G-stochastic analysis and mean-field SDEs. If the aggregation point is fixed and the typos cleaned up, the result is solid. A serious referee should be able to sort it out in one round.

Referee Report

3 major / 4 minor

Summary. The paper studies the solution map of a mean-field $G$-SDE whose coefficients depend Lipschitz-continuously on the state $x$ and on the random variable $\xi \in L^{2,d}_*$. It establishes continuity and growth estimates for $(x,\xi)\mapsto (X^{t,x,\xi},X^{t,\xi})$, then proves Fréchet differentiability of $x\mapsto X^{t,x,\xi}$ and of $\xi\mapsto X^{t,\xi}$ under a $C^1$-type assumption on the coefficients, characterizing the derivatives as solutions of explicit linear $G$-SDEs. A second-order theory is developed under $C^2$-type assumptions, and the final section embeds the distribution-dependent formulation of Sun--Wu--Wu into the authors' random-variable formulation. The central objects are Propositions 4.23 and 4.24; the former identifies $D_\xi X^{t,\xi}\eta = D_x X^{t,\xi,\xi}\eta + Y^{t,\xi,\eta}$ with $Y^{t,\xi,\eta}$ solving (4.4).

Significance. If the main theorems are correct, this is the first sensitivity calculus for mean-field $G$-SDEs in the Banach-space formulation of Bollweg--Meyer-Brandis, and it goes substantially beyond the well-posedness results of [2]. The explicit $G$-SDEs for the Fréchet derivatives are concrete and potentially useful for control, numerics, and further regularity analysis. The authors also make a useful observation connecting their framework to distribution-dependent coefficients via a Lions-type lifting. The significance is therefore high, provided the gaps below are closed.

major comments (3)

[Lemma 3.10 and its uses (Cor. 4.13, Lemmas 4.12, 4.18, 4.22, Prop. 4.23)] The aggregation property of the conditional sublinear expectation is invoked without proof or citation. The paper needs to substitute a random variable $x=\xi$ inside conditional $G$-expectations of unbounded functionals, e.g., $\hat E[\sup_w \|X^{t,x,\xi}_w-X^{t,\xi}_w\|^2 \mid F_t]$ evaluated at $x=\xi$. The spaces $L^{2,d}_*$ and $H^{2,d}_*(t,T)$ are completions of bounded objects, and sublinear expectations are not continuous from below; the standard independence argument only applies directly to bounded continuous functions of the future with deterministic $x$. No theorem in [25] or [2] is cited for this extension. Since the identification $X^{t,\xi,\xi}=X^{t,\xi}$ and the derivative formula in Prop. 4.23 rest on this step, a proof or a precise reference is required.
[Abstract and Section 5] The abstract claims 'second order Fréchet differentiability in the random initial condition', but no theorem in Section 5 establishes $D^2_\xi X^{t,\xi}$ or gives its $G$-SDE. Proposition 5.4 concerns $D^2_x X^{t,x,\xi}$; Propositions 5.7 and 5.9 give the mixed derivatives $D_xD_\xi X^{t,x,\xi}\eta$ and $D_\xi D_x X^{t,x,\xi}y$. These do not imply a second derivative with respect to the random initial condition $\xi$. Either the claim must be revised to 'first-order in $\xi$ and second/mixed derivatives in $(x,\xi)$', or a genuine $D^2_\xi$ statement must be added. As written, the abstract overstates the results of Section 5.
[Lemma 3.8, proof] The proof claims, for $1\le p\le q_0$, $$\hat E\Big[\sup_{t\le w\le s}\|X^{t,x,\xi}_w-X^{t,y,\eta}_w\|^p \,\Big|\, F_t\Big] \lesssim \|x-y\|^p+\|\xi-\eta\|^p_{L^2_*} + \int_t^s \alpha_0(u)^p \hat E\Big[\sup_{t\le w\le u}\|X^{t,x,\xi}_w-X^{t,y,\eta}_w\|^2\,\Big|\, F_t\Big]du.$$ The integrand contains $\sup^2$, not $\sup^p$, so Grönwall does not close the estimate for the $p$th moment. For $p<2$ the term $\hat E[\sup^2] $ is not controlled by the desired quantity, and for $p>2$ there is a mismatch of exponents. This lemma is used repeatedly (e.g., in Prop. 4.9 and Lemma 4.10), so the proof must be corrected, presumably by replacing $\sup^2$ with $\sup^p$ and using the appropriate Grönwall argument.

minor comments (4)

[Lemma 4.2] The fundamental theorem of calculus is misstated: it reads $f(v_0+v)-f(v)=\int_0^1 Df(v_0+\lambda v)v\,d\lambda$, but the second argument should be $v_0$, i.e., $f(v_0+v)-f(v_0)$.
[Proposition 4.23] The proposition states the map $\xi\mapsto X^{t,\xi}$ goes from $L^{2,d}_*(t)$ to $H^{1,d}_*(t,T)$, but the displayed Fréchet derivative is written as $D_\xi X^{t,\xi}:L^{2,d}_*(t)\to H^{2,d}_*(t,T)$. Since the derivative as an operator into $H^1$ is the natural statement, the codomain in the boxed definition should be $H^{1,d}_*(t,T)$ (or the theorem should explicitly say the derivative has a continuous extension to $H^2$). Also the symbol $D_xX^{t,\xi}$ in the proposition should be $D_\xi X^{t,\xi}$.
[Lemma 4.12] In the proof, 'Corollary 3.9' should be 'Lemma 3.9'.
[Throughout] There are numerous small typos, e.g., 'the the', 'followin', 'A^{t,x,\xi,y}_w y' in Lemma 4.8, and inconsistent use of $D_xX^{t,\xi}$ vs. $D_\xi X^{t,\xi}$ in Proposition 4.23. These should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; differentiability results are new and derived from stated hypotheses. The unproven aggregation property is a correctness gap, not a circular reduction.

full rationale

The derivation is not circular. The paper's core results—Fréchet differentiability of ξ ↦ X^{t,ξ} (Prop. 4.23), ξ ↦ X^{t,x,ξ} (Prop. 4.24), and the second-order derivatives (Props. 5.4, 5.7, 5.9)—are proved from the stated coefficient hypotheses (Assumptions 3.1, 4.3, 5.1) via Grönwall estimates on explicitly constructed increment processes (A, Y, C, D). The derivative representations are derived, not assumed: Y^{t,ξ,η} is first defined as the unique solution of a linear G-SDE (4.4) and then shown in Lemma 4.22/Prop. 4.23 to satisfy the required Fréchet limit. No fitted parameters are involved, no quantity is renamed as a prediction, and no uniqueness theorem from the authors' prior work is used to forbid alternatives. The paper does rely on the authors' earlier article [2] for existence/uniqueness and some technical lemmas, but [2] is a published, peer-reviewed paper whose results are not the conclusions derived here; this is legitimate external support and does not constitute circularity. The only notable weakness is Lemma 3.10's unproven invocation of the 'aggregation property' to substitute a random initial condition into a conditional G-expectation, an issue that recurs in Lemmas 4.12, 4.18 and 4.22. That is a potential gap in proof—no theorem is cited for the extension to the completed spaces L^{2,d}_* and H^{2,d}_*(t,T)—but it is not a self-referential reduction. The equality X^{t,ξ,ξ}=X^{t,ξ} is not assumed as the theorem's conclusion, and the differentiability claim would not follow by construction even if that equality were granted. Accordingly, no circular step is present.

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

The paper introduces no free parameters and no invented entities. It builds directly on Peng's G-expectation theory (domain assumption), the authors' earlier well-posedness theorem for the random-variable mean-field G-SDE (domain assumption), and the aggregation property of conditional sublinear expectation (domain assumption). Standard analytic tools (Grönwall, BDG, FTC) are used without proof. The only 'assumptions' beyond the framework are the coefficient regularity hypotheses (Assumptions 3.1, 4.3, 5.1), which are stated as part of the theorems rather than hidden premises.

assumptions (5)

domain assumption G-framework: G-Brownian motion, sublinear expectation, Itô integral, BDG inequality (Peng [25], Chapter 8)
The entire mathematical setting is taken from Peng's theory; the paper does not prove these foundations.
domain assumption Well-posedness of the random-variable mean-field G-SDE (Theorem 3.12 in [2])
Proposition 3.3 and later sections rely on existence and uniqueness of (3.1), (3.2) proved in the authors' previous paper.
domain assumption Aggregation property of conditional sublinear expectation
Invoked in Lemma 3.10 and repeatedly to evaluate at x = xi; not proved here, only cited from the G-framework.
standard math Fundamental theorem of calculus for Fréchet derivatives (Lemma 4.2)
Used throughout; the statement contains a typo (f(v) instead of f(v0)) but applications use the correct form.
standard math Hölder, Grönwall and Burkholder-Davis-Gundy inequalities in the sublinear expectation framework
Used in a priori estimates; the BDG inequality under G-expectation is cited from [25].

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Cite this review

Pith. "Pith review of Regularity of Solutions of Mean-Field $G$-SDEs." pith.science (2026). https://pith.science/paper/VEAK64I7

@misc{pith2026250807867,
  author       = {Pith},
  title        = {Pith review of: Regularity of Solutions of Mean-Field $G$-SDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VEAK64I7}},
  note         = {Machine review of arXiv:2508.07867}
}

abstract

We study regularity properties of the unique solution of a mean-field $G$-SDE. More precisely, we consider a mean-field $G$-SDE with square-integrable random initial condition and establish its first and second order Fr\'echet differentiability in the random initial condition and specify the $G$-SDEs of the respective Fr\'echet derivatives.

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Works this paper leans on

34 extracted references · 28 canonical work pages

[2]

Mean-field stochastic differ- ential equations driven by G-Brownian motion

Karl-Wilhelm Georg Bollweg and Thilo Meyer-Brandis. “Mean-field stochastic differ- ential equations driven by G-Brownian motion”. In:Probability, Uncertainty and Quan- titative Risk (Apr. 2025). Publisher: Probability, Uncertainty and Quantitative Risk, pp. 1–24. issn: 2095-9672. doi: 10.3934/puqr.2025011. 45

work page doi:10.3934/puqr.2025011 2025
[25]

Nonlinear expectations and stochastic calculus under uncertainty: with Ro- bust CLT and G-Brownian motion

Shige Peng. Nonlinear expectations and stochastic calculus under uncertainty: with Ro- bust CLT and G-Brownian motion. Probability theory and stochastic modelling volume

work page
[1]

Non-linear affine pro- cesses with jumps

Francesca Biagini, Georg Bollweg, and Katharina Oberpriller. “Non-linear affine pro- cesses with jumps”. en. In:Probability, Uncertainty and Quantitative Risk8.2 (Mar. 2023), pp. 235–266.issn: 2095-9672. doi: 10.3934/puqr.2023010

work page doi:10.3934/puqr.2023010 2023
[3]

Mean-field stochastic differential equations and associated PDEs

Rainer Buckdahn et al. “Mean-field stochastic differential equations and associated PDEs”. In: (July 2014). arXiv: 1407.1215

work page Pith review arXiv 2014
[4]

Mean field game of controls and an application to trade crowding

Pierre Cardaliaguet and Charles-Albert Lehalle. “Mean field game of controls and an application to trade crowding”. en. In: Mathematics and Financial Economics 12.3 (June 2018), pp. 335–363. issn: 1862-9679, 1862-9660. doi: 10.1007/s11579- 017- 0206-z

work page doi:10.1007/s11579- 2018
[5]

Probabilistic theory of mean field games with applications I: Mean field FBSDEs, control, and games

René Carmona and Francois Delarue. Probabilistic theory of mean field games with applications I: Mean field FBSDEs, control, and games. Vol. 1st. Probability theory and stochastic modelling volume 83. 2018.isbn: 978-3-319-56437-1 978-3-319-58920-6

work page 2018
[6]

Probabilistic theory of mean field games with applications II: mean field games with common noise and master equations

René Carmona and Francois Delarue. Probabilistic theory of mean field games with applications II: mean field games with common noise and master equations. 1st edition. Vol. 2nd. New York, NY: Springer Science+Business Media, 2017.isbn: 978-3-319- 56435-7

work page 2017
[7]

Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces

Samuel Cohen. “Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces”. In: Electronic Journal of Probability 17.none (Jan. 2012). issn: 1083-6489. doi: 10.1214/EJP.v17-2224

work page doi:10.1214/ejp.v17-2224 2012

Show all 34 references

[8]

Affine processes under pa- rameter uncertainty

Tolulope Fadina, Ariel Neufeld, and Thorsten Schmidt. “Affine processes under pa- rameter uncertainty”. en. In:Probability, Uncertainty and Quantitative Risk4.1 (Dec. 2019), p. 5.issn: 2367-0126. doi: 10.1186/s41546-019-0039-1

2019 doi
[9]

Lévy-type processes under uncertainty and related nonlocal equations

Julian Hollender. Lévy-type processes under uncertainty and related nonlocal equations. eng. OCLC: 962061637. Dresden: Technische Universität Dresden, 2016.isbn: 978-1- 5355-5384-1

2016
[10]

BSDEs driven by G-Brownian motion under degenerate case and its application to the regularity of fully nonlinear PDEs

Mingshang Hu, Shaolin Ji, and Xiaojuan Li. BSDEs driven by G-Brownian motion under degenerate case and its application to the regularity of fully nonlinear PDEs. arXiv:2205.09164 [math]. May 2022.doi: 10.48550/arXiv.2205.09164

2022 doi arXiv
[11]

G-Lévyprocessesundersublinearexpectations

MingshangHuandShigePeng.“G-Lévyprocessesundersublinearexpectations”.en.In: Probability, Uncertainty and Quantitative Risk6.1 (Mar. 2021). Publisher: Probability, Uncertainty and Quantitative Risk, pp. 1–22.issn: 2095-9672. doi: 10.3934/puqr. 2021001

2021 doi
[12]

Backward stochastic differential equations driven by G-Brownian motion

Mingshang Hu et al. “Backward stochastic differential equations driven by G-Brownian motion”. en. In:Stochastic Processes and their Applications124.1 (Jan. 2014), pp. 759–

2014
[13]

Foundations of Kinetic Theory

Mark Kac. “Foundations of Kinetic Theory”. In:Proceedings of the Third Berkeley Sym- posium on Mathematical Statistics and Probability, Volume 3: Contributions to Astron- omy and Physics 3.3 (Jan. 1956). Publisher: University of California Press, pp. 171– 198

1956
[14]

Jeux à champ moyen. I – Le cas station- naire

Jean-Michel Lasry and Pierre-Louis Lions. “Jeux à champ moyen. I – Le cas station- naire”. fr. In: Comptes Rendus Mathematique 343.9 (Nov. 2006), pp. 619–625. issn: 1631-073X. doi: 10.1016/j.crma.2006.09.019. 46

2006 doi
[15]

Jeux à champ moyen. II – Horizon fini et contrôle optimal

Jean-Michel Lasry and Pierre-Louis Lions. “Jeux à champ moyen. II – Horizon fini et contrôle optimal”. fr. In:Comptes Rendus Mathematique343.10 (Nov. 2006), pp. 679–

2006
[16]

Doubly reflected backward SDEs driven by G-Brownian mo- tions and fully nonlinear PDEs with double obstacles

Hanwu Li and Ning Ning. “Doubly reflected backward SDEs driven by G-Brownian mo- tions and fully nonlinear PDEs with double obstacles”. en. In:Stochastics and Partial Differential Equations: Analysis and Computations(Mar. 2025). Company: Springer Distributor: Springer Institut...

2025 doi
[17]

A Class of Markov Processes Associated with Nonlinear Parabolic Equations

Henry P. McKean. “A Class of Markov Processes Associated with Nonlinear Parabolic Equations”. In: Proceedings of the National Academy of Sciences56.6 (Dec. 1966), pp. 1907–1911. issn: 0027-8424, 1091-6490.doi: 10.1073/pnas.56.6.1907

1966 doi
[18]

Nonlinear Lévy Processes and their Characteristics

Ariel Neufeld and Marcel Nutz. “Nonlinear Lévy Processes and their Characteristics”. en. In:Transactions of the American Mathematical Society369.1 (Mar. 2016). arXiv: 1401.7253, pp. 69–95.issn: 1088-6850. doi: 10.1090/tran/6656

2016 arXiv
[19]

Constructing sublinear expectations on path space

Marcel Nutz and Ramon van Handel. “Constructing sublinear expectations on path space”. en. In:Stochastic Processes and their Applications123.8 (Aug. 2013), pp. 3100–

2013
[20]

A New Central Limit Theorem under Sublinear Expectations

Shige Peng. “A New Central Limit Theorem under Sublinear Expectations”. In: (Mar. 2008). arXiv: 0803.2656

2008 arXiv
[21]

Filtration Consistent Nonlinear Expectations and Evaluations of Contin- gent Claims

Shige Peng. “Filtration Consistent Nonlinear Expectations and Evaluations of Contin- gent Claims”. en. In:Acta Mathematicae Applicatae Sinica, English Series20.2 (June 2004), pp. 191–214.issn: 1618-3932. doi: 10.1007/s10255-004-0161-3

2004 doi
[22]

G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô Type

Shige Peng. “G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô Type”. en. In:Stochastic Analysis and Applications. Ed. by Fred Espen Benth et al. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007, pp. 541–567.isbn: 978-3-540- 70846-9. doi: 10.1007/978-...

2007 doi
[23]

G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito’s type

Shige Peng. G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito’s type. arXiv:math/0601035 version: 2. Dec. 2006.doi: 10.48550/arXiv.math/ 0601035

2006 doi arXiv
[24]

Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

Shige Peng. “Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation”. en. In:Stochastic Processes and their Applications118.12 (Dec. 2008), pp. 2223–2253.issn: 03044149. doi: 10.1016/j.spa.2007.10.015

2008 doi
[26]

BSDE, path-dependent PDE and nonlinear Feynman- Kac formula

ShiGe Peng and FaLei Wang. “BSDE, path-dependent PDE and nonlinear Feynman- Kac formula”. en. In:Science China Mathematics 59.1 (Jan. 2016), pp. 19–36.issn: 1869-1862. doi: 10.1007/s11425-015-5086-1. 47

2016 doi
[27]

On distribution dependent stochastic differential equations driven by G-Brownian motion

DeSun,Jiang-LunWu,andPanyuWu. On distribution dependent stochastic differential equations driven by G-Brownian motion. arXiv:2302.12539 [math]. Feb. 2023

2023 arXiv
[28]

Mean-field backward stochastic differential equations driven by G- Brownian motion and related partial differential equations

Shengqiu Sun. “Mean-field backward stochastic differential equations driven by G- Brownian motion and related partial differential equations”. In:Mathematical Methods in the Applied Sciences43.12 (2020), pp. 7484–7505.issn: 1099-1476. doi: 10.1002/ mma.6573

2020
[29]

Topics in propagation of chaos

Alain-Sol Sznitman. “Topics in propagation of chaos”. In:Ecole d’Eté de Probabilités de Saint-Flour XIX — 1989. Ed. by Donald L. Burkholder et al. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer, 1991, pp. 165–251.isbn: 978-3-540-46319-

1989
[34]

doi: 10.1007/BFb0085169. 48

doi
[95]

doi: 10.1007/978- 3662-59903-7

Berlin [Heidelberg]: Springer, 2019.isbn: 978-3-662-59902-0. doi: 10.1007/978- 3662-59903-7

2019 doi
[684]

doi: 10.1016/j.crma.2006.09.018

issn: 1631-073X. doi: 10.1016/j.crma.2006.09.018

2006 doi
[784]

doi: 10.1016/j.spa.2013.09.010

issn: 0304-4149. doi: 10.1016/j.spa.2013.09.010

2013 doi
[3121]

doi: 10.1016/j.spa.2013.03.022

issn: 03044149. doi: 10.1016/j.spa.2013.03.022

2013 doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [2]

Mean-field stochastic differ- ential equations driven by G-Brownian motion

Karl-Wilhelm Georg Bollweg and Thilo Meyer-Brandis. “Mean-field stochastic differ- ential equations driven by G-Brownian motion”. In:Probability, Uncertainty and Quan- titative Risk (Apr. 2025). Publisher: Probability, Uncertainty and Quantitative Risk, pp. 1–24. issn: 2095-9672. doi: 10.3934/puqr.2025011. 45

work page doi:10.3934/puqr.2025011 2025

[2] [25]

Nonlinear expectations and stochastic calculus under uncertainty: with Ro- bust CLT and G-Brownian motion

Shige Peng. Nonlinear expectations and stochastic calculus under uncertainty: with Ro- bust CLT and G-Brownian motion. Probability theory and stochastic modelling volume

work page

[3] [1]

Non-linear affine pro- cesses with jumps

Francesca Biagini, Georg Bollweg, and Katharina Oberpriller. “Non-linear affine pro- cesses with jumps”. en. In:Probability, Uncertainty and Quantitative Risk8.2 (Mar. 2023), pp. 235–266.issn: 2095-9672. doi: 10.3934/puqr.2023010

work page doi:10.3934/puqr.2023010 2023

[4] [3]

Mean-field stochastic differential equations and associated PDEs

Rainer Buckdahn et al. “Mean-field stochastic differential equations and associated PDEs”. In: (July 2014). arXiv: 1407.1215

work page Pith review arXiv 2014

[5] [4]

Mean field game of controls and an application to trade crowding

Pierre Cardaliaguet and Charles-Albert Lehalle. “Mean field game of controls and an application to trade crowding”. en. In: Mathematics and Financial Economics 12.3 (June 2018), pp. 335–363. issn: 1862-9679, 1862-9660. doi: 10.1007/s11579- 017- 0206-z

work page doi:10.1007/s11579- 2018

[6] [5]

Probabilistic theory of mean field games with applications I: Mean field FBSDEs, control, and games

René Carmona and Francois Delarue. Probabilistic theory of mean field games with applications I: Mean field FBSDEs, control, and games. Vol. 1st. Probability theory and stochastic modelling volume 83. 2018.isbn: 978-3-319-56437-1 978-3-319-58920-6

work page 2018

[7] [6]

Probabilistic theory of mean field games with applications II: mean field games with common noise and master equations

René Carmona and Francois Delarue. Probabilistic theory of mean field games with applications II: mean field games with common noise and master equations. 1st edition. Vol. 2nd. New York, NY: Springer Science+Business Media, 2017.isbn: 978-3-319- 56435-7

work page 2017

[8] [7]

Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces

Samuel Cohen. “Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces”. In: Electronic Journal of Probability 17.none (Jan. 2012). issn: 1083-6489. doi: 10.1214/EJP.v17-2224

work page doi:10.1214/ejp.v17-2224 2012