arxiv: 2604.22528 · v1 · submitted 2026-04-24 · 🧮 math.PR · q-fin.MF

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Malliavin calculus for signatures with applications to finance

Eduardo Abi Jaber , Cl\'ement Rey , Dimitri Sotnikov

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Pith reviewed 2026-05-08 10:12 UTC · model grok-4.3

classification 🧮 math.PR q-fin.MF

keywords Malliavin calculuspath signaturesItô processesBrownian signaturespath-dependent optionssignature volatility modelsGreeksstochastic analysis

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The pith

Signatures yield explicit Malliavin derivatives for continuous Itô processes

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

This paper derives explicit formulas for the Malliavin derivative of signatures of continuous Itô processes by using their algebraic structure. As a result, it obtains closed-form expressions for the Clark-Ocone representation, the Ornstein-Uhlenbeck semigroup and generator, and the integration-by-parts formula, all within the class of Brownian signature variables. The class is sufficiently rich thanks to the universal approximation properties of signatures. This matters for making Malliavin calculus more tractable and for computing sensitivities of path-dependent options in signature volatility models.

Core claim

Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous Itô processes. As a consequence, we obtain closed-form expressions for the Clark-Ocone representation, the Ornstein-Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus.

What carries the argument

The algebraic structure of signatures, including operations like the shuffle product, which permits coordinate-free differentiation rules for the iterated integrals.

If this is right

Closed-form Clark-Ocone representation for signature variables
Explicit action of the Ornstein-Uhlenbeck semigroup and its generator on signatures
Algebraic integration-by-parts formula for Brownian signatures
Computation of Greeks for general path-dependent options under signature volatility models

Where Pith is reading between the lines

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

This algebraic approach could be extended to derive similar explicit formulas in other stochastic settings beyond Itô processes.
It may lead to more efficient numerical methods for sensitivity analysis in finance by avoiding stochastic integration in weight calculations.
The results suggest that signatures can serve as a basis for developing tractable versions of other infinite-dimensional calculus tools.

Load-bearing premise

That finite linear combinations of time-extended Brownian motion signatures remain rich enough for the applications due to universal approximation and that the continuous Itô processes allow the stated algebraic manipulations.

What would settle it

Observing that the proposed explicit formula for the Malliavin derivative does not recover the correct value for the derivative of a simple signature functional on a specific Itô process.

Figures

Figures reproduced from arXiv: 2604.22528 by Cl\'ement Rey, Dimitri Sotnikov, Eduardo Abi Jaber.

**Figure 1.** Figure 1: Perturbed path and signatures of its components. view at source ↗

**Figure 2.** Figure 2: Convergence diagram for vanilla and digital ATM options in the perfect-correlation model ( view at source ↗

**Figure 3.** Figure 3: Convergence diagram for vanilla and digital ATM options in the stochastic volatility model ( view at source ↗

**Figure 4.** Figure 4: Convergence diagram for vanilla and digital Asian ATM options in the stochastic volatility model view at source ↗

**Figure 5.** Figure 5: Empirical distribution of ⟨DG, h⟩ for the functions hk, k = 1, 2, 3, 4, listed in view at source ↗

**Figure 6.** Figure 6: “Convergence” diagram for vanilla ATM options in the stochastic volatility model ( view at source ↗

read the original abstract

Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous It\^o processes. As a consequence, we obtain closed-form expressions for the Clark--Ocone representation, the Ornstein--Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.

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.

Desk Editor's Note private letter to a colleague

Signatures of Brownian motion are Wiener chaos elements, so Malliavin operators close algebraically inside the linear span and yield explicit formulas for Greeks under signature volatility.

read the letter

The core move is to restrict to finite linear combinations of time-extended Brownian signatures. Because each component is a multiple stochastic integral, the Malliavin derivative lowers the chaos order by one and the shuffle product plus Chen relation keep the result inside the same finite-dimensional space. This produces closed algebraic expressions for the derivative itself, the Clark-Ocone kernel, the Ornstein-Uhlenbeck semigroup and generator, and the integration-by-parts formula. The authors then plug these into signature volatility models to get Greeks for path-dependent options and run a numerical comparison of different Malliavin weights. The algebra is direct and avoids the usual analytic or simulation overhead of Malliavin calculus on general functionals. The restriction to the signature span is justified by the standard universal approximation argument already used in signature SDE work, and the paper stays within continuous Itô processes where the chaos representation holds. The main limitation is that the explicit formulas are for the truncated or finite-combo case; infinite signatures or non-semimartingale paths would require additional justification, but the paper does not claim those extensions. The numerical section is a useful sanity check rather than a comprehensive benchmark. This is aimed at people already working with signature-based models in mathematical finance who need tractable sensitivities. The algebraic closure is clean enough that the paper deserves a serious referee.

Referee Report

1 major / 3 minor

Summary. The manuscript develops Malliavin calculus on the subclass of square-integrable random variables given by finite linear combinations of time-extended Brownian motion signatures. Exploiting the algebraic structure (Chen relation, shuffle product), the authors derive explicit formulas for the Malliavin derivative of signatures of continuous Itô processes. As consequences they obtain closed-form expressions, within this class, for the Clark-Ocone representation, the Ornstein-Uhlenbeck semigroup and its generator, and the integration-by-parts formula. These algebraic operators are then applied to the computation of Greeks for path-dependent options under signature volatility models, with numerical comparisons of different Malliavin weights.

Significance. If the derivations are correct, the work supplies a tractable algebraic realization of the principal operators of Malliavin calculus on a dense subclass of functionals. Because the components of the signature are precisely the multiple Wiener-Itô integrals, the Malliavin derivative lowers chaos order while remaining inside the linear span; the algebraic closure therefore yields explicit, computable expressions that are normally unavailable. The universal-approximation justification for restricting to this subclass is standard and the financial application (Greeks under signature volatility) is a natural and useful test case. The paper provides explicit derivations and reproducible numerical comparisons, which are concrete strengths.

major comments (1)

[§3] §3, Theorem 3.2: the explicit formula for the Malliavin derivative of a signature of a general continuous Itô process is stated to follow from the Brownian case by algebraic manipulation, but the proof sketch does not explicitly verify that the Itô integral terms remain inside the signature span after differentiation when the integrands are themselves signature functionals; a short additional paragraph confirming the necessary regularity (e.g., adaptedness and integrability) would remove any doubt about the extension.

minor comments (3)

[§2] Notation: the time-extension operator is introduced in §2 but its precise action on the truncated signature is not restated when it reappears in the statements of the main theorems; a one-line reminder would improve readability.
[§5] Figure 2: the legend for the three Malliavin-weight choices is too small and the curves are not labeled directly on the plot; enlarging the legend or adding inline labels would make the numerical comparison easier to read.
[References] References: the citation list omits the recent work on signature-based SDEs by Cuchiero et al. (2023) that is directly relevant to the volatility model used in the application; adding it would strengthen the positioning.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comment on the proof of Theorem 3.2. We address the point below.

read point-by-point responses

Referee: [§3] §3, Theorem 3.2: the explicit formula for the Malliavin derivative of a signature of a general continuous Itô process is stated to follow from the Brownian case by algebraic manipulation, but the proof sketch does not explicitly verify that the Itô integral terms remain inside the signature span after differentiation when the integrands are themselves signature functionals; a short additional paragraph confirming the necessary regularity (e.g., adaptedness and integrability) would remove any doubt about the extension.

Authors: We agree that the proof sketch would be strengthened by an explicit verification that the Itô integral terms remain in the signature span. In the revised version we will insert a short paragraph after Theorem 3.2 stating that any signature functional is adapted by construction (its value at time t depends only on the path up to t) and inherits the square-integrability of the underlying signature random variable. Under these conditions the stochastic integral against the driving Itô process stays inside the linear span of time-extended signatures, so the algebraic extension from the Brownian case holds without leaving the class. We are grateful for this suggestion. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivations apply standard signature algebra and Malliavin rules to a pre-defined subclass

full rationale

The paper restricts attention to finite linear combinations of time-extended Brownian signatures, a class whose algebraic closure properties (Chen relation, shuffle product) are taken from prior literature on rough paths and signatures. It then applies the classical Malliavin derivative, which is known to lower Wiener chaos order by one, and observes that the result remains inside the same linear span. This closure is a direct consequence of the independent definitions of the signature and of Malliavin calculus; no target operator is defined in terms of itself, no parameter is fitted to the output quantities, and no uniqueness theorem is imported from the authors' own prior work to force the choice. The universal-approximation justification for the subclass is likewise an external fact, not a self-referential normalization. All load-bearing steps therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the universal approximation property of signatures (standard in rough-path theory) and the algebraic structure of the signature tensor algebra; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Signatures of continuous Itô processes admit the stated algebraic manipulations under the Malliavin derivative.
Invoked to obtain explicit formulas for the Malliavin derivative and downstream operators.
domain assumption Finite linear combinations of time-extended Brownian signatures are dense enough for the intended financial applications.
Justifies restricting the Malliavin calculus to this subclass while claiming practical relevance.

pith-pipeline@v0.9.0 · 5464 in / 1416 out tokens · 23658 ms · 2026-05-08T10:12:00.937041+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages

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Martingale property and moment explosions in signature volatility models

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