On Expected Signatures and Signature Cumulants in Semimartingale Models

Peter K. Friz (), Paul P. Hager () and Nikolas Tapia ()
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Peter K. Friz: Technische Universität Berlin
Paul P. Hager: University of Vienna
Nikolas Tapia: Humboldt University Berlin

A chapter in Signature Methods in Finance, 2026, pp 381-424 from Springer

Abstract: 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.

Date: 2026
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DOI: 10.1007/978-3-031-97239-3_11

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