pith. sign in

arxiv: 2509.14529 · v2 · pith:UWWGODWHnew · submitted 2025-09-18 · 💱 q-fin.MF · math.PR

Unbiased Rough Integrators and No Free Lunch in Rough-Path-Based Market Models

Tomoyuki Ichiba , Qijin Shi This is my paper

Pith reviewed 2026-05-21 22:58 UTC · model grok-4.3

classification 💱 q-fin.MF math.PR
keywords rough pathsno free luncharbitragemarket modelscontrolled pathssignature strategiessemimartingalesstochastic calculus
0
0 comments X

The pith

Enlarging trading strategies in rough-path market models forces the price driver to collapse to a time-changed Brownian motion.

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

The paper develops a Rough Kreps-Yan theorem that ties a no-controlled-free-lunch condition to the unbiasedness of a rough integrator. It then classifies which random rough paths remain admissible once integrands are restricted to successively larger families of controlled paths. Markovian portfolios still allow Gaussian-Hermite rough paths, but signature portfolios and adaptedly scaled signature portfolios shrink the set further until only the Itô lift of Brownian motion survives, up to time reparametrization. The result shows that continuous frictionless markets built on rough paths are forced back into the classical semimartingale setting. The framework applies to α-Hölder rough paths for any α > 0.

Core claim

A Rough Kreps-Yan theorem equates No Controlled Free Lunch with unbiasedness of the rough integrator. Classifying these unbiased integrators across controlled-path classes reveals a progressive collapse: Markovian portfolios permit Gaussian-Hermite rough paths; signature portfolios narrow the set; adaptedly scaled signature portfolios reduce it to the Itô lift of standard Brownian motion up to time change. Simple strategies are absent from the admissible classes.

What carries the argument

Classification of unbiased rough integrators according to the class of controlled paths used as integrands, under the No Controlled Free Lunch arbitrage condition.

If this is right

  • Any continuous frictionless market model built on rough paths must employ a driver whose law is that of a time-changed Brownian motion once sufficiently rich signature strategies are allowed.
  • The admissible set of random rough paths shrinks monotonically as the portfolio class is enlarged from Markovian to signature to adaptedly scaled signature.
  • The theory excludes simple strategies, so the no-arbitrage constraint applies only to the richer controlled-path families.
  • The framework remains valid for arbitrarily rough α-Hölder paths with α > 0.

Where Pith is reading between the lines

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

  • Attempts to price or hedge with genuinely non-semimartingale rough paths may become arbitrage-free only after the admissible trading strategies are artificially restricted.
  • Signature-based strategies, already used in machine-learning finance, inherit the same collapse property and therefore cannot enlarge the set of arbitrage-free rough models beyond semimartingales.
  • One could test the result numerically by simulating signature trading on sample paths of fractional Brownian motion and checking whether controlled free lunches appear once the Hurst parameter deviates from 1/2.

Load-bearing premise

The classification assumes that the chosen families of controlled paths and the No Controlled Free Lunch notion together capture the relevant trading opportunities and arbitrage opportunities in the rough-path market.

What would settle it

Exhibit a non-semimartingale α-Hölder rough path that still satisfies No Controlled Free Lunch when integrands are taken from the adaptedly scaled signature class.

read the original abstract

Built to generalise classical stochastic calculus, rough path theory provides a natural and pathwise framework to model continuous non-semimartingale assets. This paper investigates the capacity of this framework to support frictionless continuous No-Free-Lunch markets \`a la Kreps-Yan. We establish a "Rough Kreps-Yan" theorem, which links a No Controlled Free Lunch (NCFL) condition to the unbiasedness of the driver of the price process as a rough integrator. The central work of this paper is a classification of these unbiased rough integrators with respect to different classes of controlled paths as integrands, under some assumptions. As the admissible strategies are enlarged from Markovian-type portfolios to signature-type and adaptedly scaled signature-type portfolios, the admissible random rough paths collapse first to Gaussian-Hermite rough paths, and ultimately to the It\^o rough path lift of a standard Brownian motion, up to a time change. Notably, simple strategies do not appear in the theory. This implies that within our framework, continuous frictionless markets based on rough path theory are inevitably constrained to the classical semimartingale paradigm, clarifying the limits of this approach. Our framework covers $\alpha-$H\"older continuous rough paths for $\alpha>0$ arbitrarily small in the tensor algebra setting.

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 develops a rough-path analogue of the Kreps-Yan theorem for frictionless continuous trading. It introduces a No Controlled Free Lunch (NCFL) condition and proves that NCFL is equivalent to the driver being an unbiased rough integrator with respect to admissible classes of controlled paths. The central result classifies such unbiased integrators: as the strategy class enlarges from Markovian-type portfolios to signature-type and then adaptedly scaled signature-type portfolios, the admissible random rough paths collapse first to Gaussian-Hermite rough paths and ultimately to the Itô rough-path lift of a time-changed Brownian motion. The framework applies to α-Hölder rough paths for arbitrarily small α > 0 in the tensor-algebra setting and concludes that rough-path market models are thereby constrained to the classical semimartingale paradigm.

Significance. If the classification and the Rough Kreps-Yan theorem hold, the work supplies a precise boundary on the use of non-semimartingale rough-path models for arbitrage-free pricing: enlarging the admissible integrand class forces the driver back to Brownian motion. This is a substantive negative result that clarifies the limits of the rough-path approach in mathematical finance and may guide future model construction. The explicit nesting of controlled-path classes and the pathwise formulation are technically interesting, though their force depends on whether the chosen strategy classes and the NCFL notion adequately represent realistic continuous trading.

major comments (2)
  1. [Main classification theorem / Rough Kreps-Yan theorem] The central classification (abstract and main theorem) asserts that unbiasedness with respect to adaptedly scaled signature-type portfolios forces the random rough path to be the Itô lift of a time-changed Brownian motion. The argument relies on algebraic properties of the signature and the precise definition of 'adaptedly scaled' lifts, yet the manuscript supplies no explicit verification steps, error estimates, or counter-example rough paths that remain unbiased for Markovian integrands but fail for the larger signature classes; this verification is load-bearing for the collapse claim, especially for α-Hölder regularity with α arbitrarily small.
  2. [Rough Kreps-Yan theorem] The Rough Kreps-Yan theorem equates NCFL to unbiasedness only for the stated nested classes of controlled paths (Markovian, signature, adaptedly scaled signature). The paper does not demonstrate that these classes exhaust the relevant integrands for absence of arbitrage in the rough-path setting, nor does it address whether NCFL remains equivalent to no-arbitrage when realistic continuous strategies fall outside controlled-path regularity; this assumption directly affects whether the collapse constrains actual market models.
minor comments (2)
  1. [Introduction / Main results] Notation for the tensor algebra and the precise definition of 'unbiased rough integrator' should be recalled or cross-referenced at the first use in the classification section to aid readers unfamiliar with the external rough-path literature.
  2. [Discussion] The statement that 'simple strategies do not appear in the theory' is important for the semimartingale conclusion; a brief remark clarifying why they are excluded from the admissible classes would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for acknowledging the substantive negative result in our classification of unbiased rough integrators. We address the two major comments point by point below and indicate the revisions we will undertake.

read point-by-point responses

  1. Referee: [Main classification theorem / Rough Kreps-Yan theorem] The central classification (abstract and main theorem) asserts that unbiasedness with respect to adaptedly scaled signature-type portfolios forces the random rough path to be the Itô lift of a time-changed Brownian motion. The argument relies on algebraic properties of the signature and the precise definition of 'adaptedly scaled' lifts, yet the manuscript supplies no explicit verification steps, error estimates, or counter-example rough paths that remain unbiased for Markovian integrands but fail for the larger signature classes; this verification is load-bearing for the collapse claim, especially for α-Hölder regularity with α arbitrarily small.

    Authors: We agree that additional explicit verification would strengthen the presentation. The proof in Sections 3–4 proceeds in steps: unbiasedness against Markovian controlled paths first forces the Gaussian-Hermite property via the shuffle algebra and finite-moment conditions on the signature; enlargement to (unscaled) signature portfolios then uses density of signature linear combinations in the controlled-path space to annihilate all non-Brownian iterated integrals; the adapted scaling step incorporates the time change while preserving the Itô lift. We will add (i) explicit remainder estimates for the signature approximation in the α-Hölder topology (valid for arbitrarily small α > 0), (ii) a concrete counter-example of a non-Gaussian rough path (e.g., a suitably lifted fractional Brownian motion with H ≠ 1/2) that satisfies unbiasedness for Markovian integrands but violates it for signature integrands, and (iii) a short appendix verifying the algebraic identities used for the adapted scaling. These additions will be included in the revised manuscript. revision: yes

  2. Referee: [Rough Kreps-Yan theorem] The Rough Kreps-Yan theorem equates NCFL to unbiasedness only for the stated nested classes of controlled paths (Markovian, signature, adaptedly scaled signature). The paper does not demonstrate that these classes exhaust the relevant integrands for absence of arbitrage in the rough-path setting, nor does it address whether NCFL remains equivalent to no-arbitrage when realistic continuous strategies fall outside controlled-path regularity; this assumption directly affects whether the collapse constrains actual market models.

    Authors: The referee correctly notes that our result is conditional on the chosen hierarchy of integrand classes. These classes are selected because the signature map provides a canonical, dense approximation to a broad family of continuous, path-dependent strategies within the rough-path framework; the nested structure (Markovian ⊂ signature ⊂ adaptedly scaled signature) is designed to capture increasing degrees of path dependence while remaining inside the controlled-path regularity required for the rough integral to be well-defined. We do not claim these classes are exhaustive of every conceivable continuous trading rule. If a strategy lies outside controlled paths, the precise equivalence between NCFL and unbiasedness would indeed require separate justification. We will revise the introduction and the discussion section to state this modeling scope explicitly and to note that the collapse result applies inside the controlled-path setting; outside it, the question of arbitrage remains open and is left for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external rough-path theory

full rationale

The paper derives a Rough Kreps-Yan theorem linking NCFL to unbiasedness of rough integrators and classifies them across nested controlled-path classes (Markovian to adaptedly scaled signatures), yielding collapse to Gaussian-Hermite paths and ultimately the Itô lift of time-changed Brownian motion. These steps rely on algebraic properties of the signature, definitions of admissible integrands, and classical no-free-lunch results, none of which are defined in terms of the paper's own outputs or fitted quantities. No self-citations are invoked as load-bearing premises for the central classification, and the framework uses external benchmarks from rough-path theory without reducing predictions to inputs by construction. The result is therefore independent of the target conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper introduces the new notion of an unbiased rough integrator and relies on standard rough-path axioms plus domain assumptions about controlled paths and the NCFL condition; no numerical free parameters are fitted.

axioms (2)
  • standard math Existence and uniqueness of rough-path lifts for α-Hölder paths with α > 0 in the tensor algebra
    Invoked to define the price process as a rough integrator for arbitrarily small α.
  • domain assumption Controlled paths form the admissible integrands for the rough integral
    Used to classify integrators under Markovian, signature, and adaptedly scaled signature strategies.
invented entities (1)
  • unbiased rough integrator no independent evidence
    purpose: Characterizes drivers that satisfy the Rough Kreps-Yan no-free-lunch condition
    New concept introduced to link NCFL to path properties; no independent falsifiable prediction outside the theorem is given.

pith-pipeline@v0.9.0 · 5762 in / 1521 out tokens · 64521 ms · 2026-05-21T22:58:29.539979+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages

  1. [1]

    \ Gérard, L A

    AG25 APACrefauthors Abi Jaber, E. \ Gérard, L A. APACrefauthors \ 2025 . Signature Volatility Models: Pricing and Hedging with Fourier Signature volatility models: Pricing and hedging with Fourier . SIAM Journal on Financial Mathematics 16 2 606–642 . APACrefDOI doi:10.1137/24m1636952 APACrefDOI

    work page doi:10.1137/24m1636952 2025

  2. [2]

    , Gérard, L A

    AGH24 APACrefauthors Abi Jaber, E. , Gérard, L A. \ Huang, Y. APACrefauthors \ 2024 . Path-Dependent Processes from Signatures . Path-Dependent Processes from Signatures . arXiv:2407.04956 [math.PR]

    work page arXiv 2024

  3. [3]

    APACrefauthors \ 2021

    All21 APACrefauthors Allan, A. APACrefauthors \ 2021 . Rough Path Theory. Rough path theory. Lecture Notes. ETH Z\"urich

    work page 2021

  4. [4]

    , Cuchiero, C

    ACLP23 APACrefauthors Allan, A. , Cuchiero, C. , Liu, C. \ Prömel, D. APACrefauthors \ 2023 . Model-free portfolio theory: A rough path approach Model-free portfolio theory: A rough path approach . Mathematical Finance 33 709-765 . APACrefDOI doi:10.1111/mafi.12376 APACrefDOI

    work page doi:10.1111/mafi.12376 2023

  5. [5]

    , Liu, C

    ALP24 APACrefauthors Allan, A. , Liu, C. \ Pr\"omel, D. APACrefauthors \ 2023 . A Càdlàg Rough Path Foundation for Robust Finance A càdlàg rough path foundation for robust finance . Finance and Stochastics . APACrefDOI doi:10.1007/s00780-023-00522-0 APACrefDOI

    work page doi:10.1007/s00780-023-00522-0 2023

  6. [6]

    APACrefauthors \ 2019

    Ana19 APACrefauthors Ananova, A. APACrefauthors \ 2019 . Pathwise Integration and functional calculus for paths with finite quadratic variation. Pathwise integration and functional calculus for paths with finite quadratic variation. PhD Thesis. Imperial College London. APACrefDOI doi:10.25560/66091 APACrefDOI

    work page doi:10.25560/66091 2019

  7. [7]

    , Bellani, C

    ABBC21 APACrefauthors Armstrong, J. , Bellani, C. , Brigo, D. \ Cass, T. APACrefauthors \ 2021 . Option pricing models without probability: a rough paths approach Option pricing models without probability: a rough paths approach . Mathematical Finance 31 1494-1521 . APACrefDOI doi:10.1111/mafi.12308 APACrefDOI

    work page doi:10.1111/mafi.12308 2021

  8. [8]

    \ Ionescu, A

    AI25 APACrefauthors Armstrong, J. \ Ionescu, A. APACrefauthors \ 2025 . Gamma Hedging and Rough Paths Gamma hedging and rough paths . Finance and Stochastics . arXiv:2309.05054 [q-fin.MF] . to appear

    work page arXiv 2025

  9. [9]

    , Lyons, T

    LNP20 APACrefauthors Arribas, I P. , Lyons, T. \ Nejad, S. APACrefauthors \ 2020 . Non-parametric Pricing and Hedging of Exotic Derivatives Non-parametric pricing and hedging of exotic derivatives . Applied Mathematical Finance 27 6 457--494 . APACrefDOI doi:10.1080/1350486X.2021.1891555 APACrefDOI

    work page doi:10.1080/1350486x.2021.1891555 2020

  10. [10]

    , Salvi, C

    ASS20 APACrefauthors Arribas, I P. , Salvi, C. \ Szpruch, L. APACrefauthors \ 2021 . Sig-SDEs model for quantitative finance Sig-SDEs model for quantitative finance . Proceedings of the First ACM International Conference on AI in Finance. Proceedings of the first acm international conference on ai in finance. APACrefDOI doi:10.1145/3383455.3422553 APACrefDOI

    work page doi:10.1145/3383455.3422553 2021

  11. [11]

    , Bayer, C

    BBFP25 APACrefauthors Bank, P. , Bayer, C. , Friz, P K. \ Pelizzari, L. APACrefauthors \ 2025 . Rough PDEs for Local Stochastic Volatility Models Rough PDEs for local stochastic volatility models . Mathematical Finance 35 3 661-681 . APACrefDOI doi:10.1111/mafi.12458 APACrefDOI

    work page doi:10.1111/mafi.12458 2025

  12. [12]

    , Friz, P

    BFG16 APACrefauthors Bayer, C. , Friz, P. \ Gatheral, J. APACrefauthors \ 2016 . Pricing under rough volatility. Pricing under rough volatility. Quantitative Finance 16 887-904 . APACrefDOI doi:10.1080/14697688.2015.1099717 APACrefDOI

    work page doi:10.1080/14697688.2015.1099717 2016

  13. [13]

    , Hager, P P

    BHRS23 APACrefauthors Bayer, C. , Hager, P P. , Riedel, S. \ Schoenmakers, J. APACrefauthors \ 2023 . Optimal stopping with signatures Optimal stopping with signatures . The Annals of Applied Probability 33 1 238 -- 273 . APACrefDOI doi:10.1214/22-AAP1814 APACrefDOI

    work page doi:10.1214/22-aap1814 2023

  14. [14]

    , Pelizzari, L

    BPS25 APACrefauthors Bayer, C. , Pelizzari, L. \ Schoenmakers, J. APACrefauthors \ 2025 . Primal and dual optimal stopping with signatures Primal and dual optimal stopping with signatures . Finance and Stochastics . APACrefDOI doi:10.1007/s00780-025-00570-8 APACrefDOI

    work page doi:10.1007/s00780-025-00570-8 2025

  15. [15]

    , Schachermayer, W

    BSV11_Good_Integrator APACrefauthors Beiglböck, M. , Schachermayer, W. \ Veliyev, B. APACrefauthors \ 2011 . A direct proof of the Bichteler – Dellacherie theorem and connections to arbitrage A direct proof of the Bichteler – Dellacherie theorem and connections to arbitrage . The Annals of Probability 39 6 . APACrefDOI doi:10.1214/10-AOP602 APACrefDOI

    work page doi:10.1214/10-aop602 2011

  16. [16]

    APACrefauthors \ 1934

    Bel34 APACrefauthors Bell, E T. APACrefauthors \ 1934 . Exponential Polynomials Exponential polynomials . Annals of Mathematics 35 2 258-277 . APACrefDOI doi:10.2307/1968431 APACrefDOI

    work page doi:10.2307/1968431 1934

  17. [17]

    , Sottinen, T

    BSV08 APACrefauthors Bender, C. , Sottinen, T. \ Valkeila, E. APACrefauthors \ 2008 . Pricing by hedging and no-arbitrage beyond semimartingales. Pricing by hedging and no-arbitrage beyond semimartingales. Finance and Stochastics 12 441-468 . APACrefDOI doi:10.1007/s00780-008-0074-8 APACrefDOI

    work page doi:10.1007/s00780-008-0074-8 2008

  18. [18]

    APACrefauthors \ 1981

    Bit81 APACrefauthors Bichteler, K. APACrefauthors \ 1981 . Stochastic Integration and L^p -Theory of Semimartingales Stochastic integration and L^p -theory of semimartingales . The Annals of Probability 9 1 49--89 . APACrefDOI doi:10.1214/aop/1176994509 APACrefDOI

    work page doi:10.1214/aop/1176994509 1981

  19. [19]

    \ Hult, H

    BH05 APACrefauthors Bj\"ork, T. \ Hult, H. APACrefauthors \ 2005 . A note on Wick products and the fractional Black-Scholes model A note on Wick products and the fractional Black-Scholes model . Finance and Stochastics 9 197-209 . APACrefDOI doi:10.1007/s00780-004-0144-5 APACrefDOI

    work page doi:10.1007/s00780-004-0144-5 2005

  20. [20]

    \ Chevyrev, I

    BC19 APACrefauthors Boedihardjo, H. \ Chevyrev, I. APACrefauthors \ 2019 . An isomorphism between branched and geometric rough paths An isomorphism between branched and geometric rough paths . Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques 55 2 1131 -- 1148 . APACrefDOI doi:10.1214/18-AIHP912 APACrefDOI

    work page doi:10.1214/18-aihp912 2019

  21. [21]

    , Ferrucci, E

    BFGJ24 APACrefauthors Bonesini, O. , Ferrucci, E. , Gasteratos, I. \ Jacquier, A. APACrefauthors \ 2024 . Rough differential equations for volatility. Rough differential equations for volatility. arXiv:2412.21192 [q-fin.MF]

    work page arXiv 2024

  22. [22]

    APACrefauthors \ 1957

    Che57 APACrefauthors Chen, K. APACrefauthors \ 1957 . Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Annals of Mathematics 65 163-178 . APACrefDOI doi:10.2307/1969671 APACrefDOI

    work page doi:10.2307/1969671 1957

  23. [23]

    APACrefauthors \ 2001

    Che01 APACrefauthors Cheridito, P. APACrefauthors \ 2001 . Mixed Fractional Brownian Motion Mixed fractional Brownian motion . Bernoulli 7 6 913--934 . APACrefDOI doi:10.2307/3318626 APACrefDOI

    work page doi:10.2307/3318626 2001

  24. [24]

    APACrefauthors \ 2003

    Che03 APACrefauthors Cheridito, P. APACrefauthors \ 2003 . Arbitrage in fractional Brownian motion models. Arbitrage in fractional Brownian motion models. Finance and Stochastics 7 533-553 . APACrefDOI doi:10.1007/s007800300101 APACrefDOI

    work page doi:10.1007/s007800300101 2003

  25. [25]

    , Kawaguchi, H

    CKM03 APACrefauthors Cheridito, P. , Kawaguchi, H. \ Maejima, M. APACrefauthors \ 2003 . Fractional Ornstein-Uhlenbeck processes. Fractional Ornstein-Uhlenbeck processes. Electronic Journal of Probability 8 3--14 . APACrefDOI doi:10.1214/EJP.v8-125 APACrefDOI

    work page doi:10.1214/ejp.v8-125 2003

  26. [26]

    APACrefauthors \ 2001

    Con01 APACrefauthors Cont, R. APACrefauthors \ 2001 . Empirical properties of asset returns: stylized facts and statistical issues Empirical properties of asset returns: stylized facts and statistical issues . Quantitative Finance 1 2 223--236 . APACrefDOI doi:10.1080/713665670 APACrefDOI

    work page doi:10.1080/713665670 2001

  27. [27]

    \ Fournié, D A

    CF10 APACrefauthors Cont, R. \ Fournié, D A. APACrefauthors \ 2010 . Change of variable formulas for non-anticipative functionals on path space Change of variable formulas for non-anticipative functionals on path space . Journal of Functional Analysis 259 4 1043-1072 . APACrefDOI doi:10.1016/j.jfa.2010.04.017 APACrefDOI

    work page doi:10.1016/j.jfa.2010.04.017 2010

  28. [28]

    \ Qian, Z

    CQ02 APACrefauthors Coutin, L. \ Qian, Z. APACrefauthors \ 2002 . Stochastic analysis, rough path analysis and fractional Brownian motions. Stochastic analysis, rough path analysis and fractional Brownian motions. Probability Theory and Related Fields 122 108-140 . APACrefDOI doi:10.1007/s004400100158 APACrefDOI

    work page doi:10.1007/s004400100158 2002

  29. [29]

    , Gazzani, G

    CGMS25 APACrefauthors Cuchiero, C. , Gazzani, G. , Möller, J. \ Svaluto-Ferro, S. APACrefauthors \ 2025 . Joint calibration to SPX and VIX options with signature-based models Joint calibration to SPX and VIX options with signature-based models . Mathematical Finance 35 1 161-213 . APACrefDOI doi:10.1111/mafi.12442 APACrefDOI

    work page doi:10.1111/mafi.12442 2025

  30. [30]

    , Gazzani, G

    CGS23 APACrefauthors Cuchiero, C. , Gazzani, G. \ Svaluto-Ferro, S. APACrefauthors \ 2023 . Signature-Based Models: Theory and Calibration Signature-based models: Theory and calibration . SIAM Journal on Financial Mathematics 14 3 910-957 . APACrefDOI doi:10.1137/22M1512338 APACrefDOI

    work page doi:10.1137/22m1512338 2023

  31. [31]

    \ Möller, J

    CM24 APACrefauthors Cuchiero, C. \ Möller, J. APACrefauthors \ 2024 . Signature Methods in Stochastic Portfolio Theory . Signature Methods in Stochastic Portfolio Theory . arXiv:2310.02322 [q-fin.MF]

    work page arXiv 2024

  32. [32]

    , Primavera, F

    CPS25 APACrefauthors Cuchiero, C. , Primavera, F. \ Svaluto-Ferro, S. APACrefauthors \ 2025 . Universal approximation theorems for continuous functions of c\`adl\`ag paths and L\'evy -type signature models Universal approximation theorems for continuous functions of c\`adl\`ag paths and L\'evy -type signature models . Finance and Stochastics 29 2 289--342...

    work page doi:10.1007/s00780-025-00557-5 2025

  33. [33]

    \ Kallianpur, G

    DK00 APACrefauthors Dasgupta, A. \ Kallianpur, G. APACrefauthors \ 2000 . Arbitrage Opportunities for a Class of Gladyshev Processes. Arbitrage opportunities for a class of Gladyshev processes. Applied Mathematics and Optimization 41 377-385 . APACrefDOI doi:10.1007/s002459911019 APACrefDOI

    work page doi:10.1007/s002459911019 2000

  34. [34]

    \ Schachermayer, W

    DS94 APACrefauthors Delbaen, F. \ Schachermayer, W. APACrefauthors \ 1994 . A general version of the fundamental theorem of asset pricing A general version of the fundamental theorem of asset pricing . Mathematische Annalen 300 1 463--520 . APACrefDOI doi:10.1007/BF01450498 APACrefDOI

    work page doi:10.1007/bf01450498 1994

  35. [35]

    \ Schachermayer, W

    DS98 APACrefauthors Delbaen, F. \ Schachermayer, W. APACrefauthors \ 1998 . The fundamental theorem of asset pricing for unbounded stochastic processes The fundamental theorem of asset pricing for unbounded stochastic processes . Mathematische Annalen 312 2 215--250 . APACrefDOI doi:10.1007/s002080050220 APACrefDOI

    work page doi:10.1007/s002080050220 1998

  36. [36]

    \ Schachermayer, W

    DS06 APACrefauthors Delbaen, F. \ Schachermayer, W. APACrefauthors \ 2006 . The Mathematics of Arbitrage The mathematics of arbitrage . Springer . APACrefDOI doi:10.1007/978-3-540-31299-4 APACrefDOI

    work page doi:10.1007/978-3-540-31299-4 2006

  37. [37]

    APACrefauthors \ 2007

    DiM07 APACrefauthors Di Matteo, T. APACrefauthors \ 2007 . Multi-scaling in finance Multi-scaling in finance . Quantitative Finance 7 1 21--36 . APACrefDOI doi:10.1080/14697680600969727 APACrefDOI

    work page doi:10.1080/14697680600969727 2007

  38. [38]

    \ Jost, J

    DJ23 APACrefauthors Duc, L H. \ Jost, J. APACrefauthors \ 2023 . How Rough Path Lifts Affect Expected Return and Volatility: A Rough Model under Transaction Cost How rough path lifts affect expected return and volatility: A rough model under transaction cost . SIAM Journal on Financial Mathematics 14 3 879-909 . APACrefDOI doi:10.1137/20M1358670 APACrefDOI

    work page doi:10.1137/20m1358670 2023

  39. [39]

    DHP00 APACrefauthors Duncan, T. , Hu, Y. \ Pasik-Duncan, B. APACrefauthors \ 2000 . Stochastic Calculus for Fractional Brownian Motion I . Theory Stochastic calculus for fractional Brownian motion I . Theory . SIAM Journal on Control and Optimization 38 2 582-612 . APACrefDOI doi:10.1137/S036301299834171X APACrefDOI

    work page doi:10.1137/s036301299834171x 2000

  40. [40]

    APACrefauthors \ 2019

    Dup19 APACrefauthors Dupire, B. APACrefauthors \ 2019 . Functional It\^o calculus Functional It\^o calculus . Quantitative Finance 19 5 721--729 . APACrefDOI doi:10.1080/14697688.2019.1575974 APACrefDOI

    work page doi:10.1080/14697688.2019.1575974 2019

  41. [41]

    \ Tissot-Daguette, V

    DT23 APACrefauthors Dupire, B. \ Tissot-Daguette, V. APACrefauthors \ 2023 . Functional Expansions . Functional Expansions . arXiv:2212.13628 [q-fin.MF]

    work page arXiv 2023

  42. [42]

    \ Van Der Hoek, J

    EV03 APACrefauthors Elliott, R. \ Van Der Hoek, J. APACrefauthors \ 2003 . A General Fractional White Noise Theory And Applications To Finance A general fractional white noise theory and applications to finance . Mathematical Finance 13 2 301-330 . APACrefDOI doi:10.1111/1467-9965.00018 APACrefDOI

    work page doi:10.1111/1467-9965.00018 2003

  43. [43]

    APACrefauthors \ 2021

    Fer21 APACrefauthors Ferrucci, E. APACrefauthors \ 2021 . Rough Path Perspectives on the It\^o - Stratonovich Dilemma. Rough path perspectives on the It\^o - Stratonovich dilemma. PhD Thesis. Imperial College London. APACrefDOI doi:10.25560/96036 APACrefDOI

    work page doi:10.25560/96036 2021

  44. [44]

    \ Hairer, M

    FH14 APACrefauthors Friz, P. \ Hairer, M. APACrefauthors \ 2014 . A course on rough paths: with an introduction to regularity structures. A course on rough paths: with an introduction to regularity structures. Springer . APACrefDOI doi:10.1007/978-3-319-08332-2 APACrefDOI

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

  45. [45]

    \ Shekhar, A

    FS17 APACrefauthors Friz, P. \ Shekhar, A. APACrefauthors \ 2017 . General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula . The Annals of Probability 45 4 2707 -- 2765 . APACrefDOI doi:10.1214/16-AOP1123 APACrefDOI

    work page doi:10.1214/16-aop1123 2017

  46. [46]

    \ Victoir, N

    FV10 APACrefauthors Friz, P. \ Victoir, N. APACrefauthors \ 2010 . Multidimensional stochastic processes as rough paths: theory and applications. Multidimensional stochastic processes as rough paths: theory and applications. Cambridge University Press . APACrefDOI doi:10.1017/CBO9780511845079 APACrefDOI

    work page doi:10.1017/cbo9780511845079 2010

  47. [47]

    \ Takano, R

    FT24 APACrefauthors Fukasawa, M. \ Takano, R. APACrefauthors \ 2024 . A partial rough path space for rough volatility A partial rough path space for rough volatility . Electronic Journal of Probability 29 1 -- 28 . APACrefDOI doi:10.1214/24-EJP1080 APACrefDOI

    work page doi:10.1214/24-ejp1080 2024

  48. [48]

    , Jaisson, T

    GJR18 APACrefauthors Gatheral, J. , Jaisson, T. \ Rosenbaum, M. APACrefauthors \ 2018 . Volatility is rough Volatility is rough . Quantitative Finance 18 933-949 . APACrefDOI doi:10.1080/14697688.2017.1393551 APACrefDOI

    work page doi:10.1080/14697688.2017.1393551 2018

  49. [49]

    APACrefauthors \ 2006

    Guasoni APACrefauthors Guasoni, P. APACrefauthors \ 2006 . No arbitrage under transaction costs, with fractional Brownian motion and beyond. No arbitrage under transaction costs, with fractional Brownian motion and beyond. Mathematical Finance 16 569-582 . APACrefDOI doi:10.1111/j.14679965.2006.00283.x APACrefDOI

    work page doi:10.1111/j.14679965.2006.00283.x 2006

  50. [50]

    APACrefauthors \ 2004

    Gub04 APACrefauthors Gubinelli, M. APACrefauthors \ 2004 . Controlling rough paths. Controlling rough paths. Journal of Functional Analysis 216 86-140 . APACrefDOI doi:10.1016/j.jfa.2004.01.002 APACrefDOI

    work page doi:10.1016/j.jfa.2004.01.002 2004

  51. [51]

    APACrefauthors \ 2010

    Gub10 APACrefauthors Gubinelli, M. APACrefauthors \ 2010 . Ramification of rough paths. Ramification of rough paths. Journal of Differential Equations 248 693-721 . APACrefDOI doi:10.1016/j.jde.2009.11.015 APACrefDOI

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

  52. [52]

    \ Kelly, D

    HK15 APACrefauthors Hairer, M. \ Kelly, D. APACrefauthors \ 2015 . Geometric versus non-geometric rough paths Geometric versus non-geometric rough paths . Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques 51 1 207 -- 251 . APACrefDOI doi:10.1214/13-AIHP564 APACrefDOI

    work page doi:10.1214/13-aihp564 2015

  53. [53]

    , Benth, F E

    HBS24 APACrefauthors Harang, F A. , Benth, F E. \ Straum, F. APACrefauthors \ 2024 . Universal approximation on non-geometric rough paths and applications to financial derivatives pricing. Universal approximation on non-geometric rough paths and applications to financial derivatives pricing. arXiv:2412.16009 [math.FA]

    work page arXiv 2024

  54. [54]

    , Littlewood, J

    HLP59 APACrefauthors Hardy, G. , Littlewood, J. \ Polya, G. APACrefauthors \ 1959 . Inequalities Inequalities . Cambridge University Press . APACrefDOI doi:10.1017/S0025557200027455 APACrefDOI

    work page doi:10.1017/s0025557200027455 1959

  55. [55]

    , Pitbladdo, R

    HPS84 APACrefauthors Harrison, J. , Pitbladdo, R. \ Schaefer, S. APACrefauthors \ 1984 . Continuous Price Processes in Frictionless Markets Have Infinite Variation Continuous price processes in frictionless markets have infinite variation . The Journal of Business 57 353-365 . APACrefDOI doi:10.1086/296268 APACrefDOI

    work page doi:10.1086/296268 1984

  56. [56]

    , Pang, G

    IPT22 APACrefauthors Ichiba, T. , Pang, G. \ Taqqu, M S. APACrefauthors \ 2022 . Path Properties of a Generalized Fractional Brownian Motion Path properties of a generalized fractional Brownian motion . Journal of Theoretical Probability 35 1 550--574 . APACrefDOI doi:10.1007/s10959-020-01066-1 APACrefDOI

    work page doi:10.1007/s10959-020-01066-1 2022

  57. [57]

    , Pang, G

    IPT25 APACrefauthors Ichiba, T. , Pang, G. \ Taqqu, M S. APACrefauthors \ 2025 . Semimartingale properties of a generalised fractional Brownian motion and its mixtures with applications in asset pricing Semimartingale properties of a generalised fractional Brownian motion and its mixtures with applications in asset pricing . Finance and Stochastics 29 3 7...

    work page doi:10.1007/s00780-025-00562-8 2025

  58. [58]

    APACrefauthors \ 2012

    Kel12 APACrefauthors Kelly, D. APACrefauthors \ 2012 . It \^o corrections in stochastic equations. It \^o corrections in stochastic equations. PhD Thesis . University of Warwick

    work page 2012

  59. [59]

    APACrefauthors \ 1998

    Lyo98 APACrefauthors Lyons, T. APACrefauthors \ 1998 . Differential equations driven by rough signals. Differential equations driven by rough signals. Revista Matemática Iberoamericana 14 215-310 . APACrefDOI doi:10.4171/RMI/240 APACrefDOI

    work page doi:10.4171/rmi/240 1998

  60. [60]

    \ Victoir, N

    LV07 APACrefauthors Lyons, T. \ Victoir, N. APACrefauthors \ 2007 . An extension theorem to rough paths. An extension theorem to rough paths. Annales de l'Institut Henri Poincaré C, Analyse non linéaire 24 835-847 . APACrefDOI doi:10.1016/j.anihpc.2006.07.004 APACrefDOI

    work page doi:10.1016/j.anihpc.2006.07.004 2007

  61. [61]

    APACrefauthors \ 1939

    Mar39 APACrefauthors Marcinkiewicz, J. APACrefauthors \ 1939 . Sur une propri\'et\'e de la loi de Gau Sur une propri\'et\'e de la loi de Gau . Mathematische Zeitschrift 44 1 612--618 . APACrefDOI doi:10.1007/BF01210677 APACrefDOI

    work page doi:10.1007/bf01210677 1939

  62. [62]

    APACrefauthors \ 1980

    Mem80 APACrefauthors Memin, J. APACrefauthors \ 1980 . Espaces de semi martingales et changement de probabilité Espaces de semi martingales et changement de probabilité . Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 52 1 9--39 . APACrefDOI doi:10.1007/BF00534184 APACrefDOI

    work page doi:10.1007/bf00534184 1980

  63. [63]

    , Nourdin, I

    NNT08 APACrefauthors Neuenkirch, A. , Nourdin, I. \ Tindel, S. APACrefauthors \ 2008 . Delay equations driven by rough paths Delay equations driven by rough paths . Electronic Journal of Probability 13 . APACrefDOI doi:10.1214/EJP.v13-575 APACrefDOI

    work page doi:10.1214/ejp.v13-575 2008

  64. [64]

    \ Tindel, S

    NT11 APACrefauthors Nualart, D. \ Tindel, S. APACrefauthors \ 2011 . A construction of the rough path above fractional Brownian motion using Volterra ’s representation. A construction of the rough path above fractional Brownian motion using Volterra ’s representation. The Annals of Probability 39 1061-1096 . APACrefDOI doi:10.1214/10-AOP578 APACrefDOI

    work page doi:10.1214/10-aop578 2011

  65. [65]

    \ Pr \"o mel, D

    PP16 APACrefauthors Perkowski, N. \ Pr \"o mel, D. APACrefauthors \ 2016 . Pathwise stochastic integrals for model-free finance Pathwise stochastic integrals for model-free finance . Bernoulli 22 4 2486 -- 2520 . APACrefDOI doi:10.3150/15-BEJ735 APACrefDOI

    work page doi:10.3150/15-bej735 2016

  66. [66]

    QX18 APACrefauthors Qian, Z. \ Xu, X. APACrefauthors \ 2025 . Rough Path Renormalization from Stratonovich to It\^o for Fractional Brownian Motion Rough Path Renormalization from Stratonovich to It\^o for Fractional Brownian Motion . Acta Mathematica Sinica, English Series . arXiv:1803.00335 [math.PR] . to appear

    work page arXiv 2025

  67. [67]

    APACrefauthors \ 1958

    Ree58 APACrefauthors Ree, R. APACrefauthors \ 1958 . Lie Elements and an Algebra Associated With Shuffles Lie elements and an algebra associated with shuffles . Annals of Mathematics 68 2 210--220 . APACrefDOI doi:10.2307/1970243 APACrefDOI

    work page doi:10.2307/1970243 1958

  68. [68]

    APACrefauthors \ 1998

    Sal98 APACrefauthors Salopek, S. APACrefauthors \ 1998 . Tolerance to arbitrage. Tolerance to arbitrage. Stochastic Processes and their Applications 76 217-230 . APACrefDOI doi:10.1016/S0304-4149(98)00025-8 APACrefDOI

    work page doi:10.1016/s0304-4149(98)00025-8 1998

  69. [69]

    \ Wolff, M

    SW99 APACrefauthors Schaefer, H. \ Wolff, M. APACrefauthors \ 1999 . Topological Vector Spaces Topological vector spaces . Springer New York, NY . APACrefDOI doi:10.1007/978-1-4612-1468-7 APACrefDOI

    work page doi:10.1007/978-1-4612-1468-7 1999

  70. [70]

    APACrefauthors \ 2010

    Unt10 APACrefauthors Unterberger, J. APACrefauthors \ 2010 . A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering. A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering. Stochastic Processes and their Applications 120 1444-1472 . AP...

    work page doi:10.1016/j.spa.2010.04.001 2010