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Malliavin calculus for signatures with applications to finance

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  • Eduardo Abi Jaber
  • Cl'ement Rey
  • Dimitri Sotnikov

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.

Suggested Citation

  • Eduardo Abi Jaber & Cl'ement Rey & Dimitri Sotnikov, 2026. "Malliavin calculus for signatures with applications to finance," Papers 2604.22528, arXiv.org.
    • Handle: RePEc:arx:papers:2604.22528
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    References listed on IDEAS

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    1. Daniel Levin & Terry Lyons & Hao Ni, 2013. "Learning from the past, predicting the statistics for the future, learning an evolving system," Papers 1309.0260, arXiv.org, revised Mar 2016.
    2. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
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