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On Expected Signatures and Signature Cumulants in Semimartingale Models

In: Signature Methods in Finance

Author

Listed:
  • Peter K. Friz

    (Technische Universität Berlin
    Weierstrass Institute)

  • Paul P. Hager

    (University of Vienna
    Weierstrass Institute)

  • Nikolas Tapia

    (Humboldt University Berlin)

Abstract

The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors (Friz et al., Unified signature cumulants and generalized Magnus expansions. In Forum of Mathematics, Sigma, vol. 10, p. e42, 2022) we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity.

Suggested Citation

  • Peter K. Friz & Paul P. Hager & Nikolas Tapia, 2026. "On Expected Signatures and Signature Cumulants in Semimartingale Models," Springer Finance, in: Christian Bayer & Goncalo dos Reis & Blanka Horvath & Harald Oberhauser (ed.), Signature Methods in Finance, pages 381-424, Springer.
    • Handle: RePEc:spr:sprfcp:978-3-031-97239-3_11
    DOI: 10.1007/978-3-031-97239-3_11
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