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.
Signature moments to characterize laws of stochastic processes
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abstract
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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How Fast Do Signatures Learn? Statistical Theory and Applications for Path Regression
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.