pith. sign in

arxiv: 2606.21049 · v1 · pith:LH4SUMVSnew · submitted 2026-06-19 · 🧮 math.PR · math.CA

Locality of rough path lifts

Ilya Chevyrev , Emilio Ferrucci This is my paper

Pith reviewed 2026-06-26 13:47 UTC · model grok-4.3

classification 🧮 math.PR math.CA
keywords rough pathslocalityLévy areafractional Brownian motionHölder pathsgeometric rough pathsstochastic liftstime translation invariance
0
0 comments X

The pith

No local homogeneous rough path lift exists for all γ-Hölder paths with γ ≤ 1/2.

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

The paper examines whether a rough path lift of a continuous path can be chosen locally, so that the lifted increment over any interval [s,t] depends only on the path values inside that interval. It proves that no such lift can also be homogeneous when the paths are merely γ-Hölder for every γ ≤ 1/2. A stronger statement rules out any bounded Lévy area that is local and either homogeneous or time-translation invariant. The authors exhibit an unbounded local time-translation-invariant Lévy area on all continuous paths, showing boundedness cannot be dropped without losing the negative result. On the stochastic side they classify all local square-integrable lifts of multi-dimensional fractional Brownian motion, obtaining non-existence for Hurst index H ≤ 1/4 and uniqueness up to stochastic translation for H > 1/4.

Core claim

Every Hölder continuous path admits a geometric rough path lift, yet no local homogeneous lift can be defined on the space of all γ-Hölder paths for γ ≤ 1/2. Equivalently, no Lévy area exists that is simultaneously bounded, free of further regularity assumptions, and either local and homogeneous or time translation invariant. An unbounded local time translation invariant bilinear Lévy area can nevertheless be constructed on every continuous path. For fractional Brownian motion, local square-integrable lifts exist only when H > 1/4 and are then stochastic translations of the canonical lift; under additional invariance requirements only the canonical lift survives except at H = 1/3.

What carries the argument

The locality condition on a rough path lift, which forces the lifted increment over [s,t] to be determined solely by the path increments inside [s,t], together with the structural requirements of homogeneity or time translation invariance.

If this is right

  • Any attempt to lift several paths simultaneously while preserving locality must either drop homogeneity or accept unbounded lifts.
  • For fractional Brownian motion with Hurst index H ≤ 1/4 no local square-integrable rough path lift exists at all.
  • When H > 1/4 every local square-integrable lift is a stochastic translation of the canonical rough path.
  • Imposing invariance under time translation, scaling and coordinate permutation selects only the canonical lift except at H = 1/3, where a one-parameter family remains.
  • Boundedness is essential for the negative results; without it a local time translation invariant Lévy area exists on all continuous paths.

Where Pith is reading between the lines

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

  • Removing the homogeneity assumption permits local but necessarily unbounded Lévy areas on every continuous path.
  • Applications that demand locality for physical or modeling reasons may have to tolerate either unbounded lifts or the loss of homogeneity.
  • The same locality-homogeneity tension is likely to appear for other stochastic processes whose sample paths are rougher than Brownian motion.
  • The classification for fractional Brownian motion suggests that very rough paths cannot admit local integrable lifts without additional randomness that breaks homogeneity.

Load-bearing premise

The lift or Lévy area must remain homogeneous or time translation invariant in addition to being local and bounded.

What would settle it

Exhibiting one bounded local homogeneous Lévy area defined on the space of all continuous paths would disprove the main non-existence theorem.

Figures

Figures reproduced from arXiv: 2606.21049 by Emilio Ferrucci, Ilya Chevyrev.

Figure 1
Figure 1. Figure 1: In the plot on the left, a local choice of area [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The first 12 Lindenmayer iterates Z 0 , Z1 , . . . , Z11 converging to the Lévy C curve. For each m ≥ 0, let Z ‹m denote the closed polygon obtained by following Z 2m at constant speed from (0, 0)to (1, 0) over the time interval [0, 1 2 ], and then following at constant speed the horizontal segment from (1, 0) back to (0, 0) over the time interval [ 1 2 , 1]. Finally, define the path Z : [0, 1] → R 2 by se… view at source ↗
read the original abstract

Every H\"older continuous path $X$ admits a geometric rough path lift $\boldsymbol{X}$ by the Lyons-Victoir extension theorem. A natural question that emerges when lifting more than one path at once is that of locality, namely whether the lift $\boldsymbol{X}_{s,t}$ only depends on the increments $X_{s,u}$, $u \in [s,t]$. We investigate the locality of rough path lifts in deterministic and stochastic settings. On the deterministic side, we show that no local, homogeneous rough path lift can be defined on $\gamma$-H\"older paths for all $\gamma\leq 1/2$. More strongly, we show that no L\'evy area can be defined which is at the same time bounded, with no further regularity assumptions, and either local and homogeneous or time translation-invariant. We moreover show that the boundedness requirement is sharp: an unbounded, local, time translation-invariant, and bilinear L\'evy area can be defined on all continuous paths. On the stochastic side, we classify all local, square-integrable rough path lifts of multi-dimensional fractional Brownian motion with Hurst parameter $H \in (0,1/2]$. For $H \leq 1/4$, we show that no such lifts exist, while for $H>1/4$, we show that all such lifts are stochastic translations of the canonical rough path. We further refine the classification by requiring invariance in law under time translation, scaling, and coordinate permutation, and show that only the canonical lift satisfies these constraints except at $H=1/3$, for which there is a one-parameter family of lifts.

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 / 3 minor

Summary. The manuscript proves that no local, homogeneous rough path lift can be defined on γ-Hölder paths for all γ ≤ 1/2. More strongly, no bounded Lévy area exists that is bounded with no further regularity assumptions and is either local and homogeneous or time translation-invariant. An unbounded, local, time translation-invariant, and bilinear Lévy area is constructed on all continuous paths. For multi-dimensional fractional Brownian motion with Hurst parameter H ∈ (0,1/2], all local square-integrable rough path lifts are classified: none exist for H ≤ 1/4, while for H > 1/4 all such lifts are stochastic translations of the canonical rough path. Under additional invariance in law under time translation, scaling, and coordinate permutation, only the canonical lift satisfies the constraints except at H=1/3, where a one-parameter family exists.

Significance. If the results hold, this work is significant for rough path theory by establishing fundamental obstructions to constructing local lifts under homogeneity or time-translation invariance for paths of low regularity (γ ≤ 1/2). The explicit separation of the homogeneous case from the merely local/time-invariant case, together with the concrete unbounded counter-example when homogeneity is dropped, demonstrates sharpness of the boundedness requirement. The complete classification of local square-integrable lifts for fBM, including the refinement under symmetry constraints, provides a precise picture that can inform applications in stochastic analysis. The paper explicitly credits the structural assumptions and exhibits the counter-example as a strength.

minor comments (3)
  1. [Introduction] The introduction could briefly recall the definition of a geometric rough path lift and the Lyons-Victoir theorem to make the locality question more self-contained for readers outside the immediate subfield.
  2. [Stochastic classification section] In the stochastic classification, the one-parameter family at H=1/3 is stated to exist but an explicit parametrization (e.g., via a scalar multiplier on the Lévy area) would improve readability.
  3. Notation for the Lévy area (e.g., the antisymmetric part of the second-level increment) is used without a dedicated preliminary subsection; a short display equation defining it would aid clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation of minor revision. The provided summary accurately reflects the main results on the non-existence of local homogeneous lifts for γ-Hölder paths with γ ≤ 1/2, the sharpness via the unbounded counter-example, and the classification of local square-integrable lifts for fractional Brownian motion. We appreciate the recognition of the work's significance for rough path theory.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents deterministic non-existence theorems for bounded local homogeneous (or time-translation-invariant) Lévy areas on γ-Hölder paths with γ ≤ 1/2, together with a stochastic classification for fBM lifts. These are established via explicit mathematical arguments that separate the homogeneous case from the merely local case and exhibit an unbounded counter-example when homogeneity is dropped. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes imported via citation appear in the derivation chain. The results are self-contained against external mathematical benchmarks and do not reduce to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results in rough path theory and properties of fractional Brownian motion; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Lyons-Victoir extension theorem provides at least one geometric rough path lift for Hölder paths
    Invoked to establish existence of some lift before asking about locality.
  • domain assumption Fractional Brownian motion is a centered Gaussian process with covariance determined by Hurst parameter H
    Used throughout the stochastic classification.

pith-pipeline@v0.9.1-grok · 5823 in / 1365 out tokens · 37028 ms · 2026-06-26T13:47:55.766176+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

216 extracted references · 79 canonical work pages

  1. [1]

    The Annals of Probability , number =

    Terry Lyons and Hao Ni , title =. The Annals of Probability , number =. 2015 , doi =

    2015

  2. [2]

    and Seiringer, Robert , title =

    Frank, Rupert L. and Seiringer, Robert , title =. Around the research of Vladimir Maz'ya. I. Function spaces , isbn =. 2010 , publisher =. doi:10.1007/978-1-4419-1341-8_6 , keywords =

    work page doi:10.1007/978-1-4419-1341-8_6 2010

  3. [3]

    Lyons, Terry , title =. Proc. R. Soc. Lond., Ser. A , issn =. 1991 , language =. doi:10.1098/rspa.1991.0017 , keywords =

    work page doi:10.1098/rspa.1991.0017 1991

  4. [4]

    1938 , language =

    Levy, Paul , title =. 1938 , language =

    1938

  5. [5]

    , TITLE =

    Hairer, M. , title =. Invent. Math. , issn =. 2014 , language =. doi:10.1007/s00222-014-0505-4 , keywords =

    work page doi:10.1007/s00222-014-0505-4 2014

  6. [6]

    , title =

    Baker, Charles R. , title =. Trans. Am. Math. Soc. , issn =. 1974 , language =. doi:10.2307/1996567 , keywords =

    work page doi:10.2307/1996567 1974

  7. [7]

    1988 , publisher =

    Barnsley, Michael , title =. 1988 , publisher =

    1988

  8. [8]

    , title =

    Faber, G. , title =. Math. Ann. , issn =. 1910 , language =. doi:10.1007/BF01456327 , url =

    work page doi:10.1007/bf01456327 1910

  9. [9]

    Fonctions continues sans d

    Ces. Fonctions continues sans d. Arch. der Math. u. Phys. (3) , volume =. 1906 , language =

    1906

  10. [10]

    Hairer, Martin , title =. Ann. Probab. , issn =. 2025 , language =. doi:10.1214/24-AOP1756 , keywords =

    work page doi:10.1214/24-aop1756 2025

  11. [11]

    and Chevyrev, I

    Bruned, Y. and Chevyrev, I. and Friz, P. K. and Prei. A rough path perspective on renormalization , fjournal =. J. Funct. Anal. , issn =. 2019 , language =. doi:10.1016/j.jfa.2019.108283 , keywords =

    work page doi:10.1016/j.jfa.2019.108283 2019

  12. [12]

    Chandra, Ajay and Chevyrev, Ilya and Hairer, Martin and Shen, Hao , TITLE =. Publ. Math. Inst. Hautes \'. 2022 , PAGES =. doi:10.1007/s10240-022-00132-0 , URL =

    work page doi:10.1007/s10240-022-00132-0 2022

  13. [13]

    Lyons, Terry and Victoir, Nicolas , TITLE =. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , FJOURNAL =. 2004 , NUMBER =. doi:10.1098/rspa.2003.1239 , URL =

    work page doi:10.1098/rspa.2003.1239 2004

  14. [14]

    Signature moments to characterize laws of stochastic processes , year =

    Chevyrev, Ilya and Oberhauser, Harald , copyright =. Signature moments to characterize laws of stochastic processes , year =

  15. [15]

    Hu, Yaozhong and Lu, Fei and Nualart, David , TITLE =. J. Funct. Anal. , FJOURNAL =. 2014 , NUMBER =. doi:10.1016/j.jfa.2013.09.024 , URL =

    work page doi:10.1016/j.jfa.2013.09.024 2014

  16. [16]

    and Shepp, L

    Hoffmann-J rgensen, J. and Shepp, L. A. and Dudley, R. M. , TITLE =. Ann. Probab. , FJOURNAL =. 1979 , NUMBER =

    1979

  17. [17]

    Broux, Lucas and Zambotti, Lorenzo , TITLE =. J. Funct. Anal. , FJOURNAL =. 2022 , NUMBER =. doi:10.1016/j.jfa.2022.109644 , URL =

    work page doi:10.1016/j.jfa.2022.109644 2022

  18. [18]

    Signature cumulants, ordered partitions, and independence of stochastic processes , year =

    Bonnier, Patric and Oberhauser, Harald , copyright =. Signature cumulants, ordered partitions, and independence of stochastic processes , year =

  19. [19]

    Stochastic Processes Appl

    Norros, Ilkka and Saksman, Eero , title =. Stochastic Processes Appl. , issn =. 2009 , language =. doi:10.1016/j.spa.2009.05.004 , keywords =

    work page doi:10.1016/j.spa.2009.05.004 2009

  20. [20]

    1969 , journal =

    Fermeture en probabilité de certains sous-espaces d’un espace L^2 : Application aux chaos de Wiener , author =. 1969 , journal =. doi:10.1007/bf00534116 , issn =

    work page doi:10.1007/bf00534116 1969

  21. [21]

    Bulletin of the London Mathematical Society , volume =

    Boedihardjo, Horatio and Diehl, Joscha and Mezzarobba, Marc and Ni, Hao , title =. Bulletin of the London Mathematical Society , volume =. doi:https://doi.org/10.1112/blms.12420 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms.12420 , year =

    work page doi:10.1112/blms.12420

  22. [22]

    Expected signature of stopped Brownian motion on d -dimensional

    Siran Li and Hao Ni , keywords =. Expected signature of stopped Brownian motion on d -dimensional. Journal of Functional Analysis , volume =. 2022 , issn =. doi:https://doi.org/10.1016/j.jfa.2022.109447 , url =

    work page doi:10.1016/j.jfa.2022.109447 2022

  23. [23]

    The Annals of Probability , number =

    Elisa Alòs and Olivier Mazet and David Nualart , title =. The Annals of Probability , number =. 2001 , doi =

    2001

  24. [24]

    Jolis, Maria , TITLE =. J. Math. Anal. Appl. , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.jmaa.2006.07.100 , URL =

    work page doi:10.1016/j.jmaa.2006.07.100 2007

  25. [25]

    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , number =

    Peter Friz and Sebastian Riedel , title =. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , number =. 2014 , doi =

    2014

  26. [26]

    Towghi, Nasser , TITLE =. JIPAM. J. Inequal. Pure Appl. Math. , FJOURNAL =. 2002 , NUMBER =

    2002

  27. [27]

    INTEGRABILITY AND TAIL ESTIMATES FOR GAUSSIAN ROUGH DIFFERENTIAL EQUATIONS , volume =

    Cass, Thomas and Litterer, Christian and Lyons, Terry , address =. INTEGRABILITY AND TAIL ESTIMATES FOR GAUSSIAN ROUGH DIFFERENTIAL EQUATIONS , volume =. The Annals of probability , keywords =

  28. [28]

    Nagoya mathematical journal , keywords =

    Stochastic Differential Equations in a Differentiable Manifold , volume =. Nagoya mathematical journal , keywords =. 1950 , author =

    1950

  29. [29]

    Problems in non-linear analysis , pages =

    Wiener integration on certain manifolds , year =. Problems in non-linear analysis , pages =

  30. [30]

    Uniqueness for the signature of a path of bounded variation and the reduced path group , volume =

    Hambly, Ben and Lyons, Terry , address =. Uniqueness for the signature of a path of bounded variation and the reduced path group , volume =. Annals of mathematics , language =

  31. [31]

    Friz and Atul Shekhar , title =

    Peter K. Friz and Atul Shekhar , title =. The Annals of Probability , number =. 2017 , doi =

    2017

  32. [32]

    The signature of a rough path: Uniqueness , volume =

    Boedihardjo, Horatio and Geng, Xi and Lyons, Terry and Yang, Danyu , copyright =. The signature of a rough path: Uniqueness , volume =. Advances in mathematics (New York. 1965) , keywords =

    1965

  33. [33]

    and Geddes, John R

    Perez Arribas, Imanol and Goodwin, Guy M. and Geddes, John R. and Lyons, Terry and Saunders, Kate E. A. , address =. A signature-based machine learning model for distinguishing bipolar disorder and borderline personality disorder , volume =. Translational psychiatry , language =

  34. [34]

    Bulletin of the American Mathematical Society , number =

    Kuo-Tsai Chen , title =. Bulletin of the American Mathematical Society , number =. 1977 , doi =

    1977

  35. [35]

    Young , title =

    Laurence C. Young , title =. Acta Mathematica , number =. 1936 , doi =

    1936

  36. [36]

    Calcul d'Ito sans probabilités , url =

    F. Calcul d'Ito sans probabilités , url =. Séminaire de probabilités de Strasbourg , keywords =

  37. [37]

    Proceedings of the Imperial Academy , number =

    Kiyosi It. Proceedings of the Imperial Academy , number =. 1944 , doi =

    1944

  38. [38]

    Proceedings of the Japan Academy , number =

    Kiyosi It. Proceedings of the Japan Academy , number =. 1946 , doi =

    1946

  39. [39]

    Nagoya Mathematical Journal , number =

    Kiyosi It. Nagoya Mathematical Journal , number =. 1951 , doi =

    1951

  40. [40]

    The Annals of Mathematical Statistics , number =

    Eugene Wong and Moshe Zakai , title =. The Annals of Mathematical Statistics , number =. 1965 , doi =

    1965

  41. [41]

    1965 , issn =

    On the relation between ordinary and stochastic differential equations , journal =. 1965 , issn =. doi:https://doi.org/10.1016/0020-7225(65)90045-5 , url =

    work page doi:10.1016/0020-7225(65)90045-5 1965

  42. [42]

    SIAM journal on control , language =

    A New Representation for Stochastic Integrals and Equations , volume =. SIAM journal on control , language =. 1966 , author =

    1966

  43. [43]

    Signatures of Gaussian processes and SLE curves , year =

    Boedihardjo, Horatio , school =. Signatures of Gaussian processes and SLE curves , year =

  44. [44]

    Quasi-martingales and stochastic integrals , school =

    Donald Fisk , year =. Quasi-martingales and stochastic integrals , school =

  45. [45]

    Rough path perspectives on the

    Emilio Rossi Ferrucci , year =. Rough path perspectives on the

  46. [46]

    2002 , PAGES =

    Hatcher, Allen , TITLE =. 2002 , PAGES =

    2002

  47. [47]

    Applications of Mathematics, vol

    Protter, Philip E. , TITLE =. 2005 , PAGES =. doi:10.1007/978-3-662-10061-5 , URL =

    work page doi:10.1007/978-3-662-10061-5 2005

  48. [48]

    and Victoir, Nicolas B

    Friz, Peter K. and Victoir, Nicolas B. , TITLE =. 2010 , PAGES =. doi:10.1017/CBO9780511845079 , URL =

    work page doi:10.1017/cbo9780511845079 2010

  49. [49]

    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , number =

    Peter Friz and Nicolas Victoir , title =. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , number =. 2010 , doi =

    2010

  50. [50]

    and Hairer, Martin , TITLE =

    Friz, Peter K. and Hairer, Martin , TITLE =. 2014 , PAGES =

    2014

  51. [51]

    Proceedings of the

    Malliavin, Paul , TITLE =. Proceedings of the. 1978 , MRCLASS =

    1978

  52. [52]

    Hypoelliptic second order differential equations , volume =

    H. Hypoelliptic second order differential equations , JOURNAL =. 1967 , PAGES =. doi:10.1007/BF02392081 , URL =

    work page doi:10.1007/bf02392081 1967

  53. [53]

    Hairer, Martin , TITLE =. Bull. Sci. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.bulsci.2011.07.007 , URL =

    work page doi:10.1016/j.bulsci.2011.07.007 2011

  54. [54]

    Gangolli, Ramesh , TITLE =. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete , VOLUME =. 1964 , PAGES =. doi:10.1007/BF00533608 , URL =

    work page doi:10.1007/bf00533608 1964

  55. [55]

    Florchinger, Patrick , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 1990 , NUMBER =. doi:10.1007/BF01474642 , URL =

    work page doi:10.1007/bf01474642 1990

  56. [56]

    , TITLE =

    Ramanan, S. , TITLE =. 2005 , PAGES =

    2005

  57. [57]

    Belopolskaya, Ya. I. and Dalecky, Yu. L. , Publisher =. Stochastic Equations and Differential Geometry , Year =

  58. [58]

    , Publisher =

    Gliklikh, Yuri E. , Publisher =. Global and Stochastic Analysis with Applications to Mathematical Physics , Year =

  59. [59]

    Armstrong, John and Brigo, Damiano , Date-Added =

  60. [60]

    Optimal Approximation of

    Armstrong, John and Brigo, Damiano , Note =. Optimal Approximation of

  61. [61]

    Expected signature of Gaussian processes with strictly regular kernels , Volume =

    Boedihardjo, Horatio and Papavasiliou, Anastasia and Qian, Zhongmin , Note =. Expected signature of Gaussian processes with strictly regular kernels , Volume =

  62. [62]

    Nicolaescu, Liviu , Note =. Random

  63. [63]

    Riemannian geometry and geometric analysis , Year =

    Jost, J\"urgen , Edition =. Riemannian geometry and geometric analysis , Year =

  64. [64]

    Lee, John M. , Doi =. Riemannian manifolds , Url =. 1997 , Bdsk-Url-1 =

    1997

  65. [65]

    Stochastic calculus in manifolds , Url =

    \'Emery, Michel , Doi =. Stochastic calculus in manifolds , Url =. 1989 , Bdsk-Url-1 =

    1989

  66. [66]

    and Litterer, Christian , Doi =

    Cass, Thomas and Driver, Bruce K. and Litterer, Christian , Doi =. Constrained rough paths , Url =. Proc. Lond. Math. Soc. (3) , Mrclass =. 2015 , Bdsk-Url-1 =

    2015

  67. [67]

    Rogers, L. C. G. and Williams, David , Doi =. Diffusions,. 2000 , Bdsk-Url-1 =

    2000

  68. [68]

    and Platen, Eckhard , Doi =

    Kloeden, Peter E. and Platen, Eckhard , Doi =. Numerical solution of stochastic differential equations , Url =. 1992 , Bdsk-Url-1 =

    1992

  69. [69]

    Intrinsic stochastic differential equations as jets , Url =

    Armstrong, John and Brigo, Damiano , Doi =. Intrinsic stochastic differential equations as jets , Url =. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , Number =. 2018 , Bdsk-Url-1 =. http://rspa.royalsocietypublishing.org/content/474/2210/20170559.full.pdf , Issn =

    2018

  70. [70]

    Riemannian geometry , Volume =

    Petersen, Peter , Edition =. Riemannian geometry , Volume =

  71. [71]

    Interest rate dynamics and consistent forward rate curves , Url =

    Bj\"ork, Tomas and Christensen, Bent Jesper , Doi =. Interest rate dynamics and consistent forward rate curves , Url =. Math. Finance , Mrclass =. 1999 , Bdsk-Url-1 =

    1999

  72. [72]

    Probability theory and related fields , language =

    A version of Hörmander’s theorem for the fractional Brownian motion , volume =. Probability theory and related fields , language =. 2007 , author =

    2007

  73. [73]

    2009 , PAGES =

    Calin, Ovidiu and Chang, Der-Chen , TITLE =. 2009 , PAGES =. doi:10.1017/CBO9781139195966 , URL =

    work page doi:10.1017/cbo9781139195966 2009

  74. [74]

    Gubinelli, Massimiliano , TITLE =. J. Differential Equations , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.jde.2009.11.015 , URL =

    work page doi:10.1016/j.jde.2009.11.015 2010

  75. [75]

    Carbery, Anthony and Wright, James , title =. Math. Res. Lett. , issn =. 2001 , language =. doi:10.4310/MRL.2001.v8.n3.a1 , keywords =

    work page doi:10.4310/mrl.2001.v8.n3.a1 2001

  76. [76]

    Electron

    Nourdin, Ivan and Nualart, David and Poly, Guillaume , title =. Electron. J. Probab. , issn =. 2013 , language =. doi:10.1214/EJP.v18-2181 , keywords =

    work page doi:10.1214/ejp.v18-2181 2013

  77. [77]

    and Paycha, Sylvie and Prei

    Bellingeri, Carlo and Friz, Peter K. and Paycha, Sylvie and Prei. Smooth rough paths, their geometry and algebraic renormalization , fjournal =. Vietnam J. Math. , issn =. 2022 , language =. doi:10.1007/s10013-022-00570-7 , keywords =

    work page doi:10.1007/s10013-022-00570-7 2022

  78. [78]

    Acta Appl

    Diehl, Joscha and Reizenstein, Jeremy , title =. Acta Appl. Math. , issn =. 2019 , language =. doi:10.1007/s10440-018-00227-z , keywords =

    work page doi:10.1007/s10440-018-00227-z 2019

  79. [79]

    Friz, Peter and Gassiat, Paul and Lyons, Terry , title =. Trans. Am. Math. Soc. , issn =. 2015 , language =. doi:10.1090/S0002-9947-2015-06272-2 , keywords =

    work page doi:10.1090/s0002-9947-2015-06272-2 2015

  80. [80]

    Cass, Thomas and Friz, Peter , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2010 , NUMBER =. doi:10.4007/annals.2010.171.2115 , URL =

    work page doi:10.4007/annals.2010.171.2115 2010

Showing first 80 references.