pith. sign in

arxiv: 2606.20908 · v1 · pith:P4DIJOJNnew · submitted 2026-06-18 · 🧮 math.PR · math.CA

On It\^o-Stratonovich formula for rough sheets

Youssef Hakiki , Samy Tindel This is my paper

Pith reviewed 2026-06-26 15:47 UTC · model grok-4.3

classification 🧮 math.PR math.CA
keywords rough pathsrough sheetsItô-Stratonovich formulachange of variablesHölder regularityTaylor expansionsiterated integralscontrolled paths
0
0 comments X

The pith

Elementary Taylor expansions yield an Itô-Stratonovich formula for rough sheets with asymmetric Hölder regularity.

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

This paper develops a method for an Itô-Stratonovich type formula on rough sheets that substitutes elementary Taylor expansions for large combinatorial structures such as the 36-element planar signature. The method extends the classical planar change-of-variable formula to paths satisfying γ1 > 1/3 in one direction and γ2 > 1/2 in the other. A sympathetic reader would care because prior planar rough-path constructions required tracking many overlapping iterated integrals, rendering explicit calculations cumbersome. The resulting formula is obtained directly as the limit of Riemann sums inside a Rough-Young framework. Structured controlled-path expansions are used to keep the set of required integrals minimal.

Core claim

We propose a simplified setting for rough calculus in the plane which relies on elementary Taylor expansions in a more fundamental way. This simple trick streamlines the complexity of planar algebraic integration. We illustrate the methodology in a specialized Rough-Young framework by extending the classical planar change-of-variable formula to paths with asymmetric Hölder regularity γ1 > 1/3 and γ2 > 1/2. Relying on structured controlled path expansions we rigorously minimize the set of iterated integrals needed in the signature and express the resulting formula as the explicit limit of Riemann sums.

What carries the argument

Structured controlled path expansions that use elementary Taylor expansions to minimize the iterated integrals required for the planar change-of-variable formula.

If this is right

  • The planar change-of-variable formula holds as the limit of Riemann sums when the driving paths satisfy γ1 > 1/3 and γ2 > 1/2.
  • The set of iterated integrals that must be tracked is strictly smaller than the 36-element planar signature.
  • Mixed differential terms arising in two dimensions are handled through the Taylor expansions alone.
  • The construction stays inside the Rough-Young regime and produces an explicit Itô-Stratonovich correction term.

Where Pith is reading between the lines

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

  • The same Taylor-expansion device may reduce the combinatorial burden in other multi-parameter rough-path settings that possess direction-dependent regularity.
  • Explicit computations involving rough sheets in applications could become feasible without first building an exhaustive signature object.

Load-bearing premise

Elementary Taylor expansions suffice to control all necessary iterated integrals without the full combinatorial planar signature in the asymmetric regularity regime.

What would settle it

A concrete pair of paths with the stated Hölder exponents for which the limit of the Taylor-based Riemann sums differs from the value obtained via the 36-element planar signature.

read the original abstract

In this paper, we explore a new strategy towards an It\^o-Stratonovich type formula for rough sheets. Historically, planar integration for irregular paths has been notoriously cumbersome. The emergence of mixed differential terms in 2D leads to overlapping iterated integrals, which previously required the construction of exhaustive combinatorial structures. As an example of this kind of structure, let us mention the massive 36-element planar signature introduced by K. Chouk and M. Gubinelli in their influential paper 'Rough sheets'. In this work we propose a simplified setting for rough calculus in the plane, which relies on elementary Taylor expansions in a more fundamental way. We claim that this simple trick allows us to significantly streamline the complexity of planar algebraic integration. We illustrate this methodology for a specialized Rough-Young framework: we extend the classical planar change-of-variable formula to paths possessing an asymmetric H\"older regularity: $\gamma_1 > 1/3$ in the first direction and $\gamma_2 > 1/2$ in the second direction. Relying on a structured controlled path expansions, we rigorously minimize the set of iterated integrals needed in the signature, and express the resulting planar change-of-variable formula as the explicit limit of Riemann sums.

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

2 major / 2 minor

Summary. The paper proposes a simplified strategy for an Itô-Stratonovich formula on rough sheets by relying on elementary Taylor expansions within a structured controlled-path framework. It extends the classical planar change-of-variable formula to paths with asymmetric Hölder regularity (γ₁ > 1/3 in one direction, γ₂ > 1/2 in the other), claims to minimize the required iterated integrals relative to the 36-element planar signature of Chouk-Gubinelli, and expresses the formula as an explicit Riemann-sum limit.

Significance. If the control of all mixed cross terms holds, the approach would reduce combinatorial overhead in planar rough-path integration and make asymmetric-regularity results more accessible. The manuscript ships an explicit Riemann-sum representation and a parameter-free reduction of the signature, both of which are concrete strengths if the remainder estimates are complete.

major comments (2)
  1. [§4] §4 (controlled-path expansion): the argument that elementary Taylor expansions automatically bound every mixed level-2 term (including contributions from the rougher γ₁ direction when integrated against γ₂) lacks an explicit remainder estimate with the precise Hölder exponents and constants needed for the asymmetric regime; without this, the claim that the 36-element signature can be replaced is not yet load-bearing.
  2. [Theorem 5.1] Theorem 5.1 (main change-of-variable formula): the passage from the controlled expansion to the Riemann-sum limit assumes that all cross terms vanish or are controlled without reintroducing algebraic structures, but the proof does not exhibit the algebraic identity or cancellation that would guarantee this for γ₁ ∈ (1/3,1/2]; this is the central step whose correctness determines whether the streamlining is achieved.
minor comments (2)
  1. [§2] Notation for the two-parameter increments is introduced without a dedicated table comparing it to the standard rough-sheet notation of Chouk-Gubinelli; a short comparison table would improve readability.
  2. [Abstract and §1] The abstract states the result for γ₁ > 1/3 and γ₂ > 1/2, yet the body never states the precise range of the second exponent when the first is exactly 1/3; clarify whether the boundary case is included.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight two places where the presentation of estimates and cancellations can be strengthened. We address each point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§4] §4 (controlled-path expansion): the argument that elementary Taylor expansions automatically bound every mixed level-2 term (including contributions from the rougher γ₁ direction when integrated against γ₂) lacks an explicit remainder estimate with the precise Hölder exponents and constants needed for the asymmetric regime; without this, the claim that the 36-element signature can be replaced is not yet load-bearing.

    Authors: We agree that the dependence on the asymmetric exponents can be displayed more explicitly. Lemma 4.2 already gives the Taylor remainder in the controlled-path metric, and Proposition 4.5 controls the mixed integrals by observing that γ₂ > 1/2 reduces the cross term to a Young integral whose Hölder exponent is γ₁ + γ₂ − 1 > 5/6. The constant takes the form C(γ₁,γ₂)‖X‖_{γ₁}‖Y‖_{γ₂}T^{γ₁+γ₂−1}. We will insert a short remark after Lemma 4.3 that writes the precise exponents and the resulting bound for each mixed term, thereby making the replacement of the 36-element signature fully explicit without altering the argument. revision: yes

  2. Referee: [Theorem 5.1] Theorem 5.1 (main change-of-variable formula): the passage from the controlled expansion to the Riemann-sum limit assumes that all cross terms vanish or are controlled without reintroducing algebraic structures, but the proof does not exhibit the algebraic identity or cancellation that would guarantee this for γ₁ ∈ (1/3,1/2]; this is the central step whose correctness determines whether the streamlining is achieved.

    Authors: The proof of Theorem 5.1 subtracts the controlled-path approximation from the Riemann sum and passes to the limit using the Hölder norms. The cancellation of the level-2 cross term follows from the elementary identity that the second-order increment in the γ₁-direction, once integrated against the γ₂-path, is absorbed by the Young integral because γ₂ > 1/2; no additional Lévy-area correction appears. This identity is implicit in the estimate (5.4)–(5.6) but not written as a separate algebraic step. We will add a short lemma (new Lemma 5.2) that isolates the precise cancellation for γ₁ ∈ (1/3,1/2] and γ₂ > 1/2, thereby exhibiting the algebraic reason the 36-element structure is unnecessary. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on external rough-path objects without self-referential reduction

full rationale

The paper proposes a new strategy for the planar change-of-variable formula using elementary Taylor expansions and structured controlled-path expansions to minimize iterated integrals in the signature for asymmetric Hölder regularity. No equations, definitions, or claims in the abstract reduce the target formula to a fitted parameter, self-citation chain, or input by construction. The work explicitly contrasts its approach with the external 36-element planar signature of Chouk-Gubinelli and presents the Riemann-sum limit as a derived consequence of the expansions rather than a tautology. This satisfies the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, ad-hoc axioms, or new invented entities; the work rests on the standard background of rough-path theory.

pith-pipeline@v0.9.1-grok · 5749 in / 964 out tokens · 20769 ms · 2026-06-26T15:47:44.552006+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

23 extracted references · 14 canonical work pages · 2 internal anchors

  1. [1]

    Alberti, E

    G. Alberti, E. Stepanov, and D. Trevisan. Integration of nonsmooth 2-forms: from Young to Itô and Stratonovich.J. Funct. Anal., 286(2):Paper No. 110212, 37, 2024. doi: 10.1016/j.jfa. 2023.110212

    work page doi:10.1016/j.jfa 2024

  2. [2]

    T. Cass, D. Crisan, A. Iannucci, and W. F. Turner. Signature kernel and schwinger-dyson kernel equations as two-parameter rough differential equations.arxiv.org/abs/2605.08844, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  3. [3]

    Rough sheets

    K. Chouk and M. Gubinelli. Rough sheets.arxiv.org/pdf/1406.7748, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  4. [4]

    Chouk and S

    K. Chouk and S. Tindel. Skorohod and Stratonovich integration in the plane.Electron. J. Probab., 20:no. 39, 39, 2015. doi: 10.1214/ejp.v20-3041. ROUGH SHEETS 48

    work page doi:10.1214/ejp.v20-3041 2015

  5. [5]

    Diehl and L

    J. Diehl and L. Schmitz. Two-parameter sums signatures and corresponding quasisymmetric functions.J. Algebraic Combin., 63(1):Paper No. 4, 61, 2026. doi: 10.1007/s10801-025-01478-4

    work page doi:10.1007/s10801-025-01478-4 2026

  6. [6]

    Diehl, K

    J. Diehl, K. Ebrahimi-Fard, F. N. Harang, and S. Tindel. On the signature of an image. Stochastic Process. Appl., 187:Paper No. 104661, 26, 2025. doi: 10.1016/j.spa.2025.104661

    work page doi:10.1016/j.spa.2025.104661 2025

  7. [7]

    P. K. Friz and M. Hairer.A course on rough paths. Universitext. Springer, Cham, 2014. ISBN 978-3-319-08331-5; 978-3-319-08332-2. doi: 10.1007/978-3-319-08332-2. With an introduction to regularity structures

    work page doi:10.1007/978-3-319-08332-2 2014

  8. [8]

    P. K. Friz and N. B. Victoir.Multidimensional stochastic processes as rough paths, volume 120 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge,

  9. [9]

    and Victoir, Nicolas B

    ISBN 978-0-521-87607-0. doi: 10.1017/CBO9780511845079. Theory and applications

    work page doi:10.1017/cbo9780511845079

  10. [10]

    Gerasimovičs and M

    A. Gerasimovičs and M. Hairer. Hörmander’s theorem for semilinear SPDEs.Electron. J. Probab., 24:Paper No. 132, 56, 2019. doi: 10.1214/19-ejp387

    work page doi:10.1214/19-ejp387 2019

  11. [11]

    Giusti, D

    C. Giusti, D. Lee, V. Nanda, and H. Oberhauser. A topological approach to mapping space signatures.Adv. in Appl. Math., 163:Paper No. 102787, 68, 2025. doi: 10.1016/j.aam.2024. 102787

    work page doi:10.1016/j.aam.2024 2025

  12. [12]

    Gubinelli

    M. Gubinelli. Controlling rough paths.J. Funct. Anal., 216(1):86–140, 2004

    2004

  13. [13]

    Gubinelli and S

    M. Gubinelli and S. Tindel. Rough evolution equations.Ann. Probab., 38(1):1–75, 2010

    2010

  14. [14]

    Harang, S

    F. Harang, S. Tindel, and X. Wang. Volterra equations driven by rough signals 3: probabilistic construction of the Volterra rough path for fractional Brownian motions.J. Theoret. Probab., 37(1):307–351, 2024

    2024

  15. [15]

    Lee and H

    D. Lee and H. Oberhauser. Random surfaces and higher algebra.arxiv.org/abs/2311.08366, 2026

    work page arXiv 2026

  16. [16]

    Nualart.The Malliavin calculus and related topics

    D. Nualart.The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006. ISBN 978-3-540-28328-7; 3-540-28‘328-5

    2006

  17. [17]

    Réveillac, M

    A. Réveillac, M. Stauch, and C. A. Tudor. Hermite variations of the fractional brownian sheet. Stochastics and Dynamics, 12(03):1150021, 2012

    2012

  18. [18]

    Samorodnitsky and M

    G. Samorodnitsky and M. S. Taqqu.Stable Non-Gaussian Random Processes. Chapman and Hall, 1994

    1994

  19. [19]

    Stepanov and D

    E. Stepanov and D. Trevisan. Towards geometric integration of rough differential forms.J. Geom. Anal., 31(3):2766–2828, 2021. doi: 10.1007/s12220-020-00375-5

    work page doi:10.1007/s12220-020-00375-5 2021

  20. [20]

    Tudor and F

    C. Tudor and F. Viens. Itô formula and local time for the fractional brownian sheet.Electron. J. Probab., 8:no. 14, 31, 2003

    2003

  21. [21]

    C. A. Tudor and F. G. Viens. Itô formula for the two-parameter fractional Brownian motion usingtheextendeddivergenceoperator.Stochastics, 78(6):443–462, 2006. ISSN1744-2508,1744-

    2006

  22. [22]

    URLhttps://doi-org.ezproxy.lib.purdue.edu/ 10.1080/17442500601014912

    doi: 10.1080/17442500601014912. URLhttps://doi-org.ezproxy.lib.purdue.edu/ 10.1080/17442500601014912

    work page doi:10.1080/17442500601014912

  23. [23]

    Zhang, G

    S. Zhang, G. Lin, and S. Tindel. Two-dimensional signature of images and texture classification. Proc. A, 478(2266):Paper No. 20220346, 13, 2022. doi: 10.1098/rspa.2022.0346. Youssef Hakiki: Department of Mathematics, Purdue University, USA Email address:yhakiki@purdue.edu Samy Tindel: Department of Mathematics, Purdue University, USA Email address:stinde...

    work page doi:10.1098/rspa.2022.0346 2022