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Signature moments to characterize laws of stochastic processes

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arxiv 1810.10971 v2 pith:KO4OVCT4 submitted 2018-10-25 math.ST math.PRstat.MLstat.TH

Ilya Chevyrev , Harald Oberhauser This is my paper · ORCID
classification math.STmath.PRstat.MLstat.TH
keywords processesstochasticlawsmomentssignatureallowscharacterizemetric
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The sequence of moments of a vector-valued random variable can characterize its law. We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments. This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic processes and study the topology it induces on the space of laws of stochastic processes. This metric can be kernelized using the signature kernel which allows to efficiently compute it. As an application, we provide a non-parametric two-sample hypothesis test for laws of stochastic processes.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 43 citations worldwide. Full citation record

  1. Expected Signature Kernels for L\'evy Rough Paths

    math.PR 2025-09 accept novelty 7.0 of 10

    Expected signature kernels of inhomogeneous Lévy processes satisfy a coupled Goursat PDE and ODE system, with exact finite-dimensional reduction in the continuous case and controlled truncation in the jump case.

  2. Learning with Expected Signatures: Theory and Applications

    stat.ML 2025-05 conditional novelty 7.0 of 10

    The paper proves consistency and asymptotic normality for empirical expected signature estimators under irregular and dependent sampling and proposes a martingale correction that lowers estimator variance.

  3. Signature Cumulants, Ordered Partitions, and Independence of Stochastic Processes

    math.PR 2019-08 accept novelty 7.0 of 10

    The logarithm of the expected path signature satisfies ordered-partition analogues of classical cumulant formulas, giving an independence test and efficient estimators for path-valued data.

  4. How Fast Do Signatures Learn? Statistical Theory and Applications for Path Regression

    math.ST 2026-07 conditional novelty 6.0 of 10

    For smooth functionals of Itô diffusions, the level-K truncated signature achieves minimax-optimal squared L2 error of order K^{-2γ}, and this rate propagates through Signature-OLS, Signature-LASSO, and Signature-Logistic.

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