pith. sign in

arxiv: 2606.27181 · v1 · pith:RFKSIWLMnew · submitted 2026-06-25 · 🧮 math.OC · cs.NA· math.NA

Numerical Approximation for Path-Dependent McKean-Vlasov Control with Non-Asymptotic Error Estimates

Olivier Bokanowski , Jean-Francois Chassagneux , Xinyu Li , Christoph Reisinger This is my paper

Pith reviewed 2026-06-26 03:20 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords path-dependent McKean-Vlasov controlEuler discretizationparticle approximationdynamic programming principlenon-asymptotic error boundsneural network policy gradientopen-loop controls
0
0 comments X

The pith

Euler discretization with piecewise-constant controls approximates path-dependent McKean-Vlasov control with non-asymptotic error O(h^{1/4}).

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

The paper develops a numerical framework for continuous-time path-dependent McKean-Vlasov control problems under open-loop controls. An Euler scheme with piecewise-constant controls is shown to converge at rate O(h^{1/4}). A discrete dynamic programming principle is proved that equates the value of open-loop controls to that of history-dependent feedback controls, allowing optimization on a reduced filtration. An interacting particle system then approximates the mean-field interaction, producing a combined error bound O(h^{1/4}) plus O(M^{-γ}) for M particles. The resulting method is implemented via a neural-network policy-gradient algorithm.

Core claim

The paper proves that an Euler discretization scheme with piecewise-constant controls achieves a non-asymptotic error of O(h^{1/4}) for path-dependent McKean-Vlasov control under open-loop controls. It establishes a discrete dynamic programming principle and proves value equivalence between open-loop and history-dependent feedback controls. An interacting particle system approximates the continuous-time value, yielding an overall error bound of O(h^{1/4}) + O(M^{-γ}) for M particles with explicitly given γ > 0. A fully implementable neural-network policy-gradient method using pathwise features is proposed and tested on a path-dependent linear-quadratic benchmark.

What carries the argument

Euler discretization scheme with piecewise-constant controls together with the discrete dynamic programming principle that equates open-loop and feedback values.

If this is right

  • Optimization reduces to a smaller filtration thanks to the open-loop/feedback equivalence.
  • The particle approximation adds an explicit rate O(M^{-γ}) that can be balanced against the time-step error.
  • A neural-network policy-gradient algorithm becomes fully implementable using pathwise features extracted from the Euler scheme.
  • Numerical tests on a path-dependent linear-quadratic example confirm practical convergence.

Where Pith is reading between the lines

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

  • The same discretization and particle rates could be tested on mean-field games with common noise to see whether the quarter-power rate persists.
  • Extending the discrete DPP to controls that depend on the full empirical measure history would allow direct comparison with closed-loop particle methods.
  • The neural-network implementation suggests that similar policy-gradient schemes might work for other path-dependent mean-field problems once the reduced filtration is available.

Load-bearing premise

The coefficients are Lipschitz continuous with suitable growth and a discrete dynamic programming principle holds for the discretized problem.

What would settle it

A concrete path-dependent McKean-Vlasov control problem with Lipschitz coefficients where the observed approximation error exceeds O(h^{1/4}) for small h, or where open-loop and feedback values differ after discretization.

Figures

Figures reproduced from arXiv: 2606.27181 by Christoph Reisinger, Jean-Francois Chassagneux, Olivier Bokanowski, Xinyu Li.

Figure 1
Figure 1. Figure 1: Comparison between the theoretical solution and Algorithm [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Generalization test under three different initial distributions using the same trained neural [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Training loss (left) and validation loss (right) of three controls: [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
read the original abstract

Path-dependent McKean--Vlasov (MKV) control models large interacting populations with history-dependent dynamics and costs. This paper develops a unified approximation-and-learning framework for continuous time path-dependent MKV problem under open-loop controls. First, an Euler discretization scheme with piecewise-constant controls is shown to achieve a non-asymptotic error of $O(h^{1/4})$. Second, we establish a discrete dynamic programming principle and prove value equivalence between open-loop and history-dependent feedback controls, enabling optimization on a reduced filtration. Third, an interacting particle system is introduced to approximate the continuous-time value, yielding an overall error bound of $O(h^{1/4}) + O(M^{-\gamma})$ for $M$ particles and an explicitly given $\gamma > 0$. Finally, we propose a fully implementable neural-network policy-gradient method using pathwise features. Numerical experiments, including a path-dependent linear-quadratic benchmark, demonstrate the effectiveness of the algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper develops a unified approximation framework for continuous-time path-dependent McKean-Vlasov control under open-loop controls. It shows that an Euler scheme with piecewise-constant controls achieves a non-asymptotic error bound of O(h^{1/4}), establishes a discrete dynamic programming principle proving value equivalence between open-loop and history-dependent feedback controls, introduces an interacting particle system yielding an overall error of O(h^{1/4}) + O(M^{-γ}) with explicit γ > 0, and proposes a neural-network policy-gradient algorithm using pathwise features, with numerical validation on a path-dependent linear-quadratic benchmark.

Significance. If the stated error estimates and equivalence results hold under the assumed Lipschitz/growth conditions, the work supplies rigorous non-asymptotic guarantees for a class of history-dependent mean-field control problems that are otherwise intractable. The explicit rates, reduction via the discrete DPP to a lower-dimensional filtration, and fully implementable learning procedure constitute a concrete advance for both theory and computation in path-dependent MKV control.

minor comments (2)
  1. [Abstract] The abstract states that γ is 'explicitly given' yet does not display its dependence on the Lipschitz constants or dimension; adding the precise expression (or a reference to the theorem that produces it) would improve readability.
  2. The numerical section would benefit from a brief statement of the network architecture (depth, width, activation) and the precise pathwise features used, to allow direct reproduction of the reported experiments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the unified approximation framework for path-dependent McKean-Vlasov control and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes error bounds for Euler discretization (O(h^{1/4})) and particle approximation (O(M^{-γ})) of path-dependent McKean-Vlasov control via a discrete dynamic programming principle under explicit Lipschitz/growth assumptions on coefficients, followed by standard convergence analysis. These steps rely on classical stochastic control techniques and propagation-of-chaos results rather than any fitted parameters, self-definitional equations, or load-bearing self-citations that reduce the claimed rates to the inputs by construction. The value equivalence between open-loop and feedback controls is proved from the DPP under the stated regularity, with no renaming of known results or ansatz smuggling. The framework is externally falsifiable via the explicit γ and benchmark numerics, making the central claims independent of the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions for existence, uniqueness, and convergence of solutions to path-dependent McKean-Vlasov equations; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The drift, diffusion, and cost coefficients satisfy Lipschitz continuity and linear growth conditions sufficient for well-posedness of the path-dependent MKV dynamics.
    Required for the Euler scheme and particle approximation to converge at the stated rates.

pith-pipeline@v0.9.1-grok · 5717 in / 1358 out tokens · 56294 ms · 2026-06-26T03:20:03.278708+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

65 extracted references · 2 canonical work pages

  1. [1]

    The Annals of Applied Probability , number =

    Jean-Fran. The Annals of Applied Probability , number =. 2019 , doi =

    2019

  2. [2]

    Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967) , pages=

    Propagation of chaos for a class of non-linear parabolic equations , author=. Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967) , pages=

    1967

  3. [3]

    Large population stochastic dynamic games: closed-loop

    Huang, Minyi and Malham. Large population stochastic dynamic games: closed-loop. Communications in Information & Systems , volume=

  4. [4]

    arXiv preprint arXiv:2309.04317 , year=

    Actor critic learning algorithms for mean-field control with moment neural networks , author=. arXiv preprint arXiv:2309.04317 , year=

    arXiv

  5. [5]

    ESAIM: Control, Optimisation and Calculus of Variations , volume=

    Bellman equation and viscosity solutions for mean-field stochastic control problem , author=. ESAIM: Control, Optimisation and Calculus of Variations , volume=. 2018 , publisher=

    2018

  6. [6]

    Discrete time

    Pham, Huy. Discrete time. Applied Mathematics & Optimization , volume=. 2016 , publisher=

    2016

  7. [7]

    Probabilistic Theory of Mean Field Games with Applications

    Carmona, Ren. Probabilistic Theory of Mean Field Games with Applications. 2018 , publisher=

    2018

  8. [8]

    Forward-backward stochastic differential equations and controlled

    Carmona, Ren. Forward-backward stochastic differential equations and controlled. The Annals of Probability , volume=

  9. [9]

    Japanese Journal of Mathematics , volume=

    Mean field games , author=. Japanese Journal of Mathematics , volume=

  10. [10]

    2013 , publisher=

    Mean Field Games and Mean Field Type Control Theory , author=. 2013 , publisher=

    2013

  11. [11]

    SIAM Journal on Control and Optimization , volume=

    Linear-quadratic optimal control problems for mean-field stochastic differential equations , author=. SIAM Journal on Control and Optimization , volume=

  12. [12]

    Ecole d'

    Topics in propagation of chaos , author=. Ecole d'. 1991 , publisher=

    1991

  13. [13]

    Proceedings of the National Academy of Sciences , volume=

    Solving high-dimensional partial differential equations using deep learning , author=. Proceedings of the National Academy of Sciences , volume=

  14. [14]

    Communications in Mathematics and Statistics , volume=

    Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations , author=. Communications in Mathematics and Statistics , volume=

  15. [15]

    arXiv preprint arXiv:2503.17869 , year=

    Learning algorithms for mean field optimal control , author=. arXiv preprint arXiv:2503.17869 , year=

    arXiv

  16. [16]

    arXiv preprint arXiv:2509.00904 , year=

    Convergence Rates of Time Discretization in Extended Mean Field Control , author=. arXiv preprint arXiv:2509.00904 , year=

    arXiv

  17. [17]

    Electronic Communications of Probability , volume =

    Improved order 1/4 convergence for piecewise constant policy approximation of stochastic control problems , author=. Electronic Communications of Probability , volume =

  18. [18]

    Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies , journal =

    Krylov, N , year=. Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies , journal =

  19. [19]

    Optimal control of path-dependent

    Cosso, Andrea and Gozzi, Fausto and Kharroubi, Idris and Pham, Huy. Optimal control of path-dependent. The Annals of Applied Probability , volume=. 2023 , publisher=

    2023

  20. [20]

    Mathematics and Computers in Simulation , volume=

    A discrete stochastic Gronwall lemma , author=. Mathematics and Computers in Simulation , volume=. 2018 , publisher=

    2018

  21. [21]

    Probability and Stochastic Modelling

    Stochastic optimal control in infinite dimension , author=. Probability and Stochastic Modelling. Springer , year=

  22. [22]

    2008 , publisher=

    Optimal transport: old and new , author=. 2008 , publisher=

    2008

  23. [23]

    On the rate of convergence in

    Fournier, Nicolas and Guillin, Arnaud , journal=. On the rate of convergence in. 2015 , publisher=

    2015

  24. [24]

    Zero-sum stochastic differential games of generalized

    Cosso, Andrea and Pham, Huy. Zero-sum stochastic differential games of generalized. Journal de Math. 2019 , publisher=

    2019

  25. [25]

    arXiv preprint arXiv:2503.20510 , year=

    Extended mean field control: a finite-dimensional numerical approximation , author=. arXiv preprint arXiv:2503.20510 , year=

    arXiv

  26. [26]

    1996 , publisher=

    Stochastic optimal control: the discrete-time case , author=. 1996 , publisher=

    1996

  27. [27]

    Mean-field neural networks-based algorithms for

    Pham, Huy. Mean-field neural networks-based algorithms for. arXiv preprint arXiv:2212.11518 , year=

    arXiv

  28. [28]

    2002 , isbn =

    Olav Kallenberg , title =. 2002 , isbn =

    2002

  29. [29]

    2017 , publisher=

    Random measures, theory and applications , author=. 2017 , publisher=

    2017

  30. [30]

    A. F. Filippov , title =. SIAM Journal on Control , year =

  31. [31]

    Computers & Mathematics with Applications , volume =

    Olivier Bokanowski and Nidhal Gammoudi and Hasnaa Zidani , title =. Computers & Mathematics with Applications , volume =. 2022 , doi =

    2022

  32. [32]

    2005 , publisher =

    Martingale Methods in Financial Modelling , author =. 2005 , publisher =

    2005

  33. [33]

    1998 , publisher =

    Methods of Mathematical Finance , author =. 1998 , publisher =

    1998

  34. [34]

    Journal of Risk , volume =

    Optimal execution of portfolio transactions , author =. Journal of Risk , volume =

  35. [35]

    1993 , publisher =

    Introduction to Functional Differential Equations , author =. 1993 , publisher =

    1993

  36. [36]

    Theory of Optimal Search , author =

  37. [37]

    2008 , publisher =

    Mathematical Epidemiology , editor =. 2008 , publisher =

    2008

  38. [38]

    Compactification methods in the control of degenerate diffusions:

    El Karoui, Nicole and Du' H. Compactification methods in the control of degenerate diffusions:. Stochastics , issn =. 1987 , language =. doi:10.1080/17442508708833443 , keywords =

    work page doi:10.1080/17442508708833443 1987

  39. [39]

    Journal of Scientific Computing , volume=

    Germain, Maximilien and Lauri. Journal of Scientific Computing , volume=. 2022 , publisher=

    2022

  40. [40]

    Numerical resolution of

    Germain, Maximilien and Mikael, Joseph and Warin, Xavier , journal=. Numerical resolution of. 2022 , publisher=

    2022

  41. [41]

    Learning high-dimensional

    Han, Jiequn and Hu, Ruimeng and Long, Jihao , journal=. Learning high-dimensional. 2024 , publisher=

    2024

  42. [42]

    arXiv preprint arXiv:2306.04788 , year=

    Deep learning for population-dependent controls in mean field control problems with common noise , author=. arXiv preprint arXiv:2306.04788 , year=

    arXiv

  43. [43]

    A fast iterative

    Reisinger, Christoph and Stockinger, Wolfgang and Zhang, Yufei , journal=. A fast iterative. 2024 , publisher=

    2024

  44. [44]

    Dayanikli and M

    G. Dayanikli and M. Lauri. Deep Learning for Population-Dependent Controls in Mean Field Control Problems with Common Noise , booktitle =

  45. [45]

    Frontiers in Applied Mathematics and Statistics , volume=

    Deep learning methods for mean field control problems with delay , author=. Frontiers in Applied Mathematics and Statistics , volume=. 2020 , publisher=

    2020

  46. [46]

    The Annals of Applied Probability , volume=

    CONVERGENCE ANALYSIS OF MACHINE LEARNING ALGORITHMS FOR THE NUMERICAL SOLUTION OF MEAN FIELD CONTROL AND GAMES , author=. The Annals of Applied Probability , volume=. 2022 , publisher=

    2022

  47. [47]

    Journal of Machine Learning Research , volume=

    Actor-critic learning for mean-field control in continuous time , author=. Journal of Machine Learning Research , volume=

  48. [48]

    arXiv preprint arXiv:2601.11217 , year=

    Model-free policy gradient for discrete-time mean-field control , author=. arXiv preprint arXiv:2601.11217 , year=

    arXiv

  49. [49]

    arXiv preprint arXiv:2107.04568 , volume=

    Deep learning for mean field games and mean field control with applications to finance , author=. arXiv preprint arXiv:2107.04568 , volume=

    arXiv

  50. [50]

    Electronic Journal of Probability , volume=

    Extended mean field control problem: a propagation of chaos result , author=. Electronic Journal of Probability , volume=. 2022 , publisher=

    2022

  51. [51]

    Communications in Mathematical Sciences , volume=

    Mean field games and systemic risk , author=. Communications in Mathematical Sciences , volume=. 2015 , doi=

    2015

  52. [52]

    SIAM Journal on Applied Mathematics , volume=

    Controlling propagation of epidemics via mean-field control , author=. SIAM Journal on Applied Mathematics , volume=. 2021 , doi=

    2021

  53. [53]

    Bioinspiration & Biomimetics , volume=

    Mean-field models in swarm robotics: a survey , author=. Bioinspiration & Biomimetics , volume=. 2020 , doi=

    2020

  54. [54]

    Contributions to Partial Differential Equations and Applications , series=

    Mean field games for modeling crowd motion , author=. Contributions to Partial Differential Equations and Applications , series=. 2019 , doi=

    2019

  55. [55]

    ESAIM: Control, Optimisation and Calculus of Variations , volume=

    Nash equilibria for a model of traffic flow with several groups of drivers , author=. ESAIM: Control, Optimisation and Calculus of Variations , volume=. 2012 , doi=

    2012

  56. [56]

    Quantitative Finance , volume=

    Volatility is rough , author=. Quantitative Finance , volume=. 2018 , doi=

    2018

  57. [57]

    Quantitative Finance , volume=

    Deep hedging , author=. Quantitative Finance , volume=. 2019 , doi=

    2019

  58. [58]

    Management Science , volume=

    Path-dependent options: Extending the Monte Carlo simulation approach , author=. Management Science , volume=. 1997 , publisher=

    1997

  59. [59]

    Vecer, Jan and Xu, Mingxin , journal=. Pricing. 2003 , publisher=

    2003

  60. [60]

    The Journal of Finance , volume=

    Path dependent options: The case of lookback options , author=. The Journal of Finance , volume=. 1991 , publisher=

    1991

  61. [61]

    Mean-field controls with

    Gu, Haotian and Guo, Xin and Wei, Xiaoli and Xu, Renyuan , journal=. Mean-field controls with. 2021 , publisher=

    2021

  62. [62]

    arXiv preprint arXiv:2605.10313 , year=

    Signature Approach for Contextual Bandits with Nonlinear and Path-dependent Rewards , author=. arXiv preprint arXiv:2605.10313 , year=

    Pith/arXiv arXiv

  63. [63]

    Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages=

    Transportation marketplace rate forecast using signature transform , author=. Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages=

  64. [64]

    2023 , issn =

    Itô’s formula for flows of measures on semimartingales , journal =. 2023 , issn =. doi:https://doi.org/10.1016/j.spa.2023.02.004 , url =

    work page doi:10.1016/j.spa.2023.02.004 2023

  65. [65]

    2009 , publisher=

    Stochastic optimal control problems for pension funds management , author=. 2009 , publisher=

    2009